Cat 6 Cable

Hello, i am fairly new to this site and i have a few questions about networking.

Setting: a home network with a business class cable connection running about 20-30 Mbps download and 4-6 Mbps upload according to speedtest.net with 1gb LAN on the router and all switches

1. would using cat 6 cable from the modem to the router increase speed and/or reduce latency on the rest of the network?

2. would running cat 6 to a server on the network have any increasing performance for users outside of the network?

3. would running a minecraft server that has at most 10 people online at a time use up so much bandwith that it would slow down everyone else on the network?

No, use cat 5e -- it is cheaper and will perform the same for your use.

With your Internet bandwidth I don't think that you will see any significant deterioration of your network performance by running a small game server.