Is it easy to mount my Time Capsule after a clean install (without any restore) so that I can copy individual files from it?
All the below answers assume that you are using TimeMachine for your backups on your TimeCapsule.

1) Will my new, clean and fresh Mountain Lion installation be able to recognize and mount Time Capsule on Finder?

Yes

2) Will I be able to easily locate, browse and copy the files from my last backup to my computer?

Yes - you will be able to find whole directory structure in folders with various time-date stamps.

3) Will I have any permission problems to copy the files?

No

4) Will my new computer try to use Time Capsule to make backups and wipe out my previous backup? (that would be very bad!)

Your new install may ask about using TimeCapsule for TimeMachine backups.  If you have a new hard drive, it will start a fresh structure.  If you have the same hard drive, it will probably continue with the same backup you currently have. But it will not wipe out previous backups, beyond the normal default backup intervals (saving once an hour for 24 hrs, once a day for a month, once a week after that).