I believe the official tf2 wiki lists them all, and that page is updated very quickly when new items are added... so I have a feeling you'd have a very hard time finding a recipe which isn't there.
Also, in case you missed it, there's a very convenient "Cost summary table" on that page, which shows you the total cost for achiving a certain item (e.g. you can see that crafting a weapon "costs" 8 other weapons).
Pretty sure by default you don't need to do anything special.
TF2 is set, via steam, to auto-update when an update is available. You can turn this off, however, at which point the game will run, but you will be unable to join any servers running on newer versions.
Similarly, if you update before servers do, you wont be able to join them, however this is rarely the case unless you catch an update right as it happens.
TF2 updates do not generally require a steam restart, nor any other individual game for that matter. Steam platform updates, however, may require a restart and generally encompass a much broader spectrum of updates (like the Source engine). It may affect TF2, but again this is automated.
Edit:
With the additional edits to the original question, I'm not sure that my initial response sufficiently answers the question. I would reference Powerlord's answer and say that when in doubt, check the official forums for patch notes, updates, and troubleshooting.
Best Answer
The item drop system gives you a random item after a certain amount of time.
Each class also has items that are awarded based on completing achievements with that class in addition to being randomly found.