Does restoring from Time Machine also restore installed applications?
Yes - if you haven't excluded the /Applications folder from the Time Machine backups. 

As for system patches - you're installing a new OS. Once you finish, load up the Software Update and it'll pull down the most recent Combo Update for you. Actually, it's better to use the combo update than a collection of patches you would have otherwise.

For more on this go to System Preferences &#x2192; Time Machine &#x2192; Options. It'll tell you what, if anything, you've excluded.
No. As long as you have sufficient space, you'll be fine. It doesn't need to be an identical drive.