Team Fortress 2 – What to Do After Completing Contracts?

As nimcap answered, unusual hats are just hats with special particle effects. They have no stat benefits, only visual effects.

The only way to obtain them is to trade for them or to find one in a Mann Co Supply Crate. Crates have a low chance to contain an unusual hat.

According to data gathered by users on the forums, it seems that series 1 crates have about 2.45% chance of containing an unusual item, and series 2 crates have about a 1.5% chance. You can tell which series of crate you have by reading the tooltip.

To open the crate you need to buy a Mann Co. Supply Crate Key, which costs $2.49 in the store. If you'd rather not spend money you can trade the crate to someone. Items that come out of crates are craftable and tradeable.

You can find more details about the drop rate on the TF2 wiki page about crates. Keep in mind that this data is gathered by players and might not be totally accurate, but it's probably in the ballpark.

You can use the simple server browser by checking the "Simple server browser" option in the bottom left hand corner. This is kind of broken for me right now, however:

You can also use the full server browser and filter by map. This is generally more reliable, but you need to know the full map name (e.g. cp_fastlane; just "fastlane" won't do):

Most servers, however, rotate through multiple maps. This is perfectly fine and a good thing. Servers that play the same map 24/7 are more likely to have bots to help people who want a game on that map regardless of time and map, and are more likely to be part of a larger server farm that makes one such server for pretty much every single map out there. Those are hardly conducive to community building.

With no community, you don't have regulars. With no regulars, you don't have donators. Without donators, you don't get money to help offset your hosting costs.

To contradict with the above, however, if you really do positively need to play on a map and there's no such server out there... you can start your own! It isn't rocket science. From the main menu click on the '+' button, pick 'cp_fastlane' or whatever as your map, tweak the settings in the Game map if you need to, make sure you have proper port forwarding in place so people can connect to your server if you're behind a router and - hey presto! You're playing on Fastlane and now you just need... others to join you. Invite your friends or something. :)