Q:

Does closing the lid put a MacBook Pro to sleep? Would it hurt the computer if I carry it around without shutting it down?

When I first got my MacBook Pro, it took less than 30 seconds to start up. Now, after several months, it takes about two minutes to startup and I can tell that it is not as responsive when using some applications (it was fast before). 
I am wondering if it is because I'm not shutting it down that causes the lag during startup.

A:

MacBooks are designed for close and go operation and you're not doing the system or hardware any harm by following the practice. That being said, there have been past issues with MacBooks not going to sleep properly when the lid is closed—the indicator being that the Mac is hot to the touch and the battery level is lower than you would expect when you remove it from your bag. There are some reports of wake from sleep issues, generally caused by a "stuck" process. 
Assuming you have adequate RAM, the slowdown in your startup process is most likely caused by applications and processes being loaded at startup. The startup on my own MacBook Pro (Mid 2007 model) is measured in minutes as various mission critical utilities—QuickSilver, TextExpander, Growl, etc.—load and populate the menubar If you login to an alternate (preferably clean, stock) user account is the startup process faster?
To get a better idea about what you're loading at startup, navigate to System Preferences and open the Users and Groups pane. (Note: Users and Groups is titled Accounts on some systems.) Select your user account from the panel on the left, then open the Login Items tab.

This screen grab is from a MacBook Air that runs only a subset of the login items of my personal MacBook Pro.
If Applications that you don't need are running, you can remove them. Do this with caution, because sometimes there are login items necessary for other software to run as expected.
Beyond that, you might want use Activity Monitor to discover what's running on your Mac. Run it right after a system startup. If you amend your question to include a screenshot of your login items and/or the running processes, we may be able to help you further.

A:

Not shutting your MacBook Pro down won't affect performance that much. There could be other factors playing a role there.
To answer the questions in your title,

By default, closing the lid on your MacBook Pro will put it to sleep. Here is some more information about sleep and other energy saving methods.
As long as your hard drive has spun down, as it does when it goes to sleep, you should be fine to put your computer to sleep and carry it around. Just make sure the sleep light starts pulsing. Once this happens, your computer has cut power to most components and has spun down the hard drive platters and parked the heads. 

Sorry about the previous answer, I had not done my research there and was going off of personal training from an individual with a strong background in Windows. Most Windows machines do not spin down the hard disks when closing the lid.
Hope this helps.

