1. The best way to contribute is by being friendly and having fun with people in the FC. You don't need to go out of your way to become some omnicrafting god, or gathering savant. If someone needs these things and you are able to help, then of course do but no one's value should come out of having to do those types of things. Just have fun talking to people and doing stuff together of you guys want to. That's how friendships are made.

2. I would say 1 week is a good length of time to determine if the FC is right for you. You will have a chance to see most active times, meet most of the members and get a feel for the weekly routine. If you hate it after 1 week, it's unlikely to get better.

3. There is, go to your GC and set one up, doesn't cost a whole lot and you only need 3/4 founding members. However, you will find recruiting to be a bit of a nightmare at first. Most players aren't going to be up for a start from scratch FC, because they won't be benefiting from it, unless you have some pretty great people already in the FC for the social aspect.

4. Talk to people. As an FC leader, I have actively recruited 2 people. My 2nd and 3rd ICs will usually do that. However, I started noticing a couple of issues with the latest recruits. They weren't being spoken to properly before being invited. We had no idea about them at all. As if they were being recruited because the FC was quiet due to being pre-patch.
One day I was playing pretty late at night, turning in some poetics for gear and i got a tell from someone, asking about my hair. We chatted for a little and went on our way. Few hours later, we bumped into each other, said hi and chatted a little more. Next night, I was on late again and to my surprise, I bumped into them again. We talked for 6 hours just idylling in Ishgard. I knew I had to invite them and they knew that we would be a good fit for them. That's how things should be to me. Inviting people for numbers and activity is pointless if you aren't going to get on. Talk to people and if you get on, see if you can join their FC, ask questions about other members and how people interact.

Hope you find somewhere you can fit in. The game is truly 100x better when playing with a good company.