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.
The easy way to do it would be to use Steam's backup feature. Select "Steam" on the menu bar, then "Backup and Restore Games...". Choose "Backup currently installed games", after which you can pick TF2 then point the backup utility at your thumbdrive to save. (It tells me it will be 6999 MB, which suggests that it compresses the backup, which may also be why some said it takes longer than just copying.) To install on other computers—restore—go to the same menu item and just pick "Restore a previous backup"
You could also just copy the team fortress 2 *.gcf (and probably the source *.gcf, and maybe the half-life 2 *.gcf's...) files out of the \steamapps folder and put them on to the computer where you want to "install" the game.
Other online sources, such as the Valve Forums, have specified these files as being the core ones:
multiplayer ob binaries.gcf
orangebox media.gcf
source 2007 binaries 2.gcf
source 2007 binaries.gcf
source 2007 shared materials.gcf
source 2007 shared models.gcf
source 2007 shared sounds.gcf
source engine.gcf
source materials.gcf
source models.gcf
source sounds.gcf
team fortress 2 client content.gcf
team fortress 2 content.gcf
team fortress 2 materials.gcf
Total size: ~11.0 GB
The files in /steamapps/[username]/team fortress 2 are reported to be 'personal' stuff, screenshots (prior to a recent update), keybindings, crash reports, etc.
You could actually copy the entire /steamapps folder, or at least all the *.gcf and *.ncf's Unless you're hard-pressed for time, I'd just use the backup/restore, as it's all but guaranteed to work.
Best Answer
The Team Fortress 2 Beta game is where the devs test changes to the game prior to rolling them out officially. It isn't intended to replace the "real" game for normal play; the rules for acquiring items are very different, for example, to facilitate testing.
More detailed information is available from the official wiki.