How to restore Mac OS X on a new hard-drive? I am going to upgrade my MacBook Pro hard-drive, but before doing so I'm trying to figure out how to restore OS X on it. My DVD drive doesn't work so I cannot re-install from the DVD.

I have a recent Time Machine backup, is that enough to restore the OS? And in that case, what would be the procedure? Also, in case the Time Machine restoration fails, is there any way to re-install OS X from a USB or external hard-drive? Any suggestion would be much appreciated.
Yes a Time Machine (TM) backup is sufficient; you don't need to "install" the OS and then restore the backup.  However you will need a copy of the OS at least and that can definitely be placed on either a USB stick or external HDD. The OS media contains the utilities you'll need to restore the TM backup.  Example here.

(edit for clarity) Depending on how new your MBP is, more specifically which version of OS X you have, you may also have a built in Recovery that has all the tools you need to do this as well.