Can I put a 7200 RPM hard drive in my MacBook Pro? I have a 2009 15" unibody MacBook Pro. It came with a 250GB, 5400 RPM hard drive. Will a 7200 RPM drive make it overheat?

I have read somewhere that putting a 7200 RPM hard drive in the MacBook Pro will make it shake and vibrate. Is this true?

Is there a certain brand that is better for the system?

And finally, is there anything else that I need to worry about?
No, no, no and no.

Have fun with the 7200 rpm drive of your choice. Mac drive mounting hardware is excellent and the drive screws are standard sized. Unless you get a drive with exotic firmware or physically too large to fit, there will be no issues relating to design. Only a faulty drive (or failing Mac) will cause issues. 

Simple physics means a 7200 RPM drive has to generate more heat than a 5400 RPM drive, but in practice, this is well within a normal Mac's ability to cool - even when you never let the drive spin down - something else like CPU load will be needed to get the internal fan to spin up.