1) You can set up Hamachi to run as a system service so that it will automatically run without having to have a user log in. Take a look at
this FAQ guide on the Hamachi forums for instructions.
2) Absolutely. As long as all the required players are a part of your Hamachi network/group and are online on Hamachi, then you can run LAN multiplayer for your games without additional configuration.
3) The VPN connection does not replace your standard Internet connection. It, in a sense, compliments it. 99% of your programs should not have any issues if they are trying to use the Internet.
If you have more questions, please do not hesitate to ask.