1. No you can't use both phones with the same number.
2. Yes, you should be able to get 3 to transfer your previous number over to your new iPhone. You could then keep the HTC as a backup and purchase a sim only package for that.
3. 3 is an official carrier for the iPhone in the UK, so no, there is no need to change if you don't want to - it depends on where you are in your HTC phone contract with them though, as to whether you are due an upgrade?
4. Syncing apps should be as easy as connecting your iPhone to your computer and syncing that way and then all the apps should be transferred.