Q:

What are the effects of orbital reflectors and advertising on scientific observation?

There's a couple of startups and initiatives which are proposing "space billboards" which have been getting a lot of bad press recently. Most of the proposals involve a bunch of cubesats with large reflective sails or what are basically large disco-balls in orbit. For example, when asked about the Humanity Star art installation, Columbia University professor David Kippin tweeted that:

This is stupid, vandalizes the night sky and corrupts our view of the cosmos

From what I've seen most of the criticism for these types of ventures fall into three categories:

People worried about a buildup of orbital debris
Scientists complaining about these types of satellites making science harder
Conservationists who want to protect the sanctity of the night sky

I'm wondering how many of these concerns are actually grounded in reality. From what I can tell, scientists aren't up in arms about Iridium flares or other satellites and the flare of the Humanity Star was pretty anticlimactic. 

Are there any documented cases of satellites, planes, or space stations ruining astronomical science? 
How long would such an orbital billboard be visible in the night sky per night?

A:

Are there any documented cases of satellites, planes, or space stations ruining astronomical science?

A similar case that comes to mind is Project Westford. In that occasion, the backlash from the scientific community started the need for scientists to participate in the planning and evaluation of outer space projects.
Sir Bernard Lovell of the Jodrell Bank Radio Observatory is quoted in the reference saying:
“The damage lies not with this experiment alone, but with the attitude of mind which makes it possible without international agreement and safeguards.”
I did not find any case of a measurement/activity that was actually disturbed on the occasion, but I do not doubt it happened. The needles had the ability to obstruct observations, specially if they reflected starlight in a fuzzy way. I would however find it a bit hard to prove in court that a corrupted measurement was caused by these needles.
See that in this case, the purpose of the experiment made sense, it only happened that the ends did not justify the means. In the case of a space billboard, the ends are not justified in themselves, according to nay-sayers to the project (myself included).
By the way, if we include amateur astronomy, this post gives a good example of Starlink satellites interfering with a long exposure picture.

How long would such an orbital billboard be visible in the night sky per night?

Depends a lot. With 3 flocks of satellites in highly-elliptical Molnya orbit, They could be visible for as long as daylight does not outshines them. 
I would expect however that these satellites would be launched in a low-earth-orbit. Then the question is how low the chosen orbit is and where they want to advertise. If they want to make an ad visible from Russia, that's different than making it visible in Equador. That's because the satellites need to have unobstructed view both to the target audience and to the Sun. 

