Lucc - yes there should be a little black valve attached to your airline. If you don't have one you can buy one very cheaply from your LFS. Just explain to them what it's for and they will show you how it works.
As for your filter slipping down well mine do that sometimes if the suction caps get a bit slimy. If you give the suction caps a good scrub to remove any slime it should help - but if that doesn't then you can always buy new suction caps. again, ask your LFS - they are also cheap to buy.
I'm sure you will get these problems sorted out soon. They are only minor things, Lucc, so try not to worry too much. In the meantime, until you get the airline and filter sorted out, you will have to just keep wiping the droplets off the inside of your hood and lighting regularly to avoid them causing mould growth.
Like I said, I had exactly the same problem but once I figured out how to set the airline flow properly, everything was fine.
If you're worried about that cut on your hand being infected, you should see your GP about it.
Regards - Athena