Q:

Could we still live if Earth's spin speed slowed down by 90%?

I am working on a project (The Longest Day), and I'm sure some people here have the necessary knowledge to give me some more usefull info.
OK, here it is:
Lets say that Earth's spinning speed diminishes to 10% of its current speed, causing days to last for 240 hours rather than 24 hours (the length of the year remains the same, the Sun is unaffected, and the Earth remains in the same orbit as it is in today). This happens gradually after a new technology is discovered and used by many countries for a long time. As usual, we abuse this new tech, and this causes the planet to slow down its spinning. Also, the unregulated use of this tech causes some wobble, thus causing mega earthquakes to happen.
Could we still survive on this planet?
How would the climate be affected? How about plant life?
Some ideas of what I have in mind:
I can see some major storms happening, the dark side having a lot of rain and being all muddy, and light side being a scorched desert after some time.
Sunrise and sundown would last very long...
Someone asked recently about the day lasting 48 hours. I'm interested in how things would be if this was heavily exagerated.

A:

We could live OK even if not well. We would sleep during long days and work during long nights - people survive in Alaska north of Polar circle. 
Agriculture would suffer significantly. Heat of long days would be mixed with cold of long nights. Many plants would have hard time to adapt to wider range of temperatures, and adapted varieties might lose some other positive qualities like taste.
It would have significant impact on climate. Summer day-storms would last for days, with adequate destruction. If some disaster happened during the night, it will be long wait to daylight to start looking for survivors. Long cold winter night might damage plants. Yes, tree can crack from cold when sap freezes.
After such big freeze, big parts of oceans might freeze, changing Earth's albedo and reflecting more sun's energy back to space (slowing warming). To fight that, we may have to release more CO2 to atmosphere to increase greenhouse effect and keep Earth warmer. Which will make summer daystorms even worse - but we may have no better options.
If things go really bad, we may need to migrate to southern/northern hemisphere according to seasons, avoiding whichever is worse: summer days or winter nights. Possibly best place to be would be in subtropical winter. Days not too hot, and nights not too cold.
Many animals on land would go extinct. Big difference might be how sudden is the change - slower change gives more time to adapt, but it will be hard either way. Insects will survive, but they will have hard time too. Birds can move long distances fast, so they will survive too, but surviving long nights without food will be challenge especially for chicks. Cannibalism between siblings might provide a chance to at least one of the baby siblings to survive first night.
Colibri might not survive 5 days night without food. Other animals which rely on sight, like bird of prey: even if they could survive without eating all long night, just born babies might not. And vice versa for animals adopted to night: bat would have to feed during the day (and face birds I guess) to survive. Owls have hard time during day: birds would attack them.
It would be less fun, but totally doable for humans.
Edit: If @Physicist137 answer is correct (and it should be with such nick, and sounds plausible), more intense radiation would be another layer of problems to deal with. Still doable but even less fun.
It is possible that most of life on dry land would be killed out in few centuries by accumulated DNA damage. Humans would have to live under protective shades at all times, and their agricultural fields had to be protected too. So far so good.
Wild animals on dry land would be extinct. Life outside of human protection will survive only in water. Humans can still harvest plants and animals from the oceans but Earth would be able to support only substantially smaller population. Surviving by scraping by. 
So, we need to get Earth to spin faster

A:

Decrease of magnetic Field: Earth shielding against deadly particles and deadly radiation coming from space, has a name: Magnetic field. The planet rotation generates a non-inertial force called "Coriolis force". The likely most accepted theory for generation of magnetic field in celestial bodies is called Dinamo Theory. Convection moves electric conducting fluid in the mantle and outer core, inside our planet. By symmetry, it is easy to note these are radially symmetric, with no preferential direction. This creates minimum magnetic field outside. However... Earth spins. This spins generates a non-inertial force called coriolis force, which organizes the convection currents into columns, generating a magnetic field aligned with the rotation axis. If earth rotation decreases, so as Coriolis force, and so as the strength of Earth's magnetic field. And then, high energy charged particles or plasma coming from the sun or interstellar winds, would reach the planet easily.
Escape of atmosphere: The constant solar wind (which is now is no more being diverted by the magnetic field) would hit molecules in Earth's atmosphere head on, transferring momentum to them. This means, more and more gas molecules of atmosphere reaches escape velocity and leaves the planet. The atmosphere would then become more and more thin until it fully vanishes. That's precisily why some large moons and even planets have no or very little atmosphere: Because they have no magnetic field. With less and less atmosphere, there will probably some unbalance in the known natural cycles of some gases (say.. nitrogen, oxygen, etc). 
Radiation: We have two shields against deadly radiation from space: Magnetic Field and atmosphere. Solar and interstellar winds and radiation could react with the Earth's  elements in the atmosphere, destroying it by making them react to produce something else, decreasing our shield strength against cosmic radiation. For example: Alpha + beta radiation can produce ozone from an oxygen molecule. People exposed to such level of radiation because atmosphere is not protecting as it should, perhaps could lead to ARS. Also, high energy charged particles (electrons, protons, positrons, ...) from solar wind, for instance, when meets a neutral atom can produce high energy photons (including gamma rays). If there is no magnetic protection to prevent this meeting in the first place, gamma radiation produced in Earth's atmosphere would increase.
Beryllium-10: Radiation (now in higher levels) might increase concentration of Beryllium-10 in Earth's atmosphere, which is an radioactive isotope and decays into beta radiation. Not sure about the consequences of this, but its probably safe to say there would be a rise in the incurable lung desease known as Berylliosis. For more information take a look here and here.
The movement of heat: A nice example to look the power of heat movement, is the Southern Ocean, at 60 degree south. The sea there is restless. The reason is the movement of heat, from the equator to poles causing: oceanic currents, persistent winds, storms, which moves over 130 million tons of water per second. Heat transfer is very powerful. Now... rotation decreased..The same that happens between equator to pole, might happen between the day and night sides. Heat will move seeking a thermalization of both sides. We are talking about storms, or worse, with unprecedental proportions, frequences and sizes. Also, we may even get powerful convective persistant winds in lower and upper atmosphere, between night and day sides.
Storms: In very simple terms... the sky is clear and sun is delivering its energy to ocean (240hrs of energy!). The water heats up. When some critical temperatures is achived, vaporization levels starts to rise significantly, causing vertical winds. Water vapor at certain altitude, releases heat and condenses to cloud, raising temperature of surrounding air, making clouds in turn rise further, producing more vertical winds. It will grow and grow and continue until the clouds block sunlight. Earth rotation forces the air to rotate forming hurricantes. However, there is no earth rotation in this case.. And as said later, there is convective persistant wind (probably, very powerful ones). This storm would be carried out by convection from the day side to the night side. Movement of this masses causes friction and thus build up of static electricity. We are talking about heavily charged clouds with tons of water inside, moving to transfer heat from day to night side. And since 240hrs the day.. its much water in this clouds....! Imagine the size... With normal Earth rotation, such events already can be seen from space.. Imagine this one.

EDIT: About the comments of the origin of the magnetic field of Earth.
This is actually a magnetohydrodynamical system explained by Dynamo Theory. The magnetic field is indeed also caused by convection. However, convection does happen because a temperature gradient between the inner core and Earth's surface. This means the fluid movement would be symetrically radial because the temperature gradient is approximately radially symmetric. And this implies no magnetic dipoles. The Coriolis force makes those convections currents be organized in columns, and then it induces the appearance of a magnetic dipole (with North pole and south pole). This means, the convection and the Coriolis force together, creates the field. There would be no magnetic dipole if there were no convection, or if there were no Coriolis force.
Without the coriolis force, the magnetic field can be approximated to the field of torus coil. And this is known to be strong only inside the coil, and not outside. Which means, the magnetic field lines would concentrate likely almost completely inside the planet.

EDIT: About the comments of Ganymede moon
Ganymede is indeed tidally locked, and indeed has a magnetic field. But this situation is beyond the scope of my answer (which dealt only with Coriolis and convection). According to this paper however, they say they can't be sure about the cause of the field, but one model is a "permanent magnetic dipole field [...] superimposed an induced magnetic dipole driven by the time varying component of the externally imposed magnetic field of Jupiter’s magnetosphere", quoting their worlds. Which means, it is really the presence of the strong external field of Jupiter which causes this particular moon to have a magnetic dipole. They of course has other models, however, the model of the moon itself generating the magnetic field is done at the order of magnetic quadrupole moment, suggesting more complications about the cause.

A:

Many plants use day length to time the cycles of their lives; experiments have been done altering these cycles, and a common consequence is that many species of plants die if exposed to unusual photoperiodicity over a long period.  Also, there would be more severe temperature swings, which would place plants under severe thermal stress that could also be lethal.
So, we'd have a world where most of the plants would be dying or dead, a recipe for mass extinction if ever I saw one.  I hardly need consider what the temperature extremes would do to animals.

