How to use the medic’s shield on Mann vs Machine

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.

Found out some information here.

Do I have to fork over 1$ every time I want to play this mode?

It seems that you do need to buy a ticket each time you want to play in Mann Up Mode, however you do not need to pay to play Mann vs. Machine.

What gives? How does Mann-Up mode work?

Playing on the Valve servers in Mann Up mode is the same as playing on regular Mann vs Machine servers, with the addition of item drops and service record information tracking, in the form of a Tour of Duty Badge.
Each successful win in Mann Up Mode gives you a exclusive Mann vs Machine item drop.
Mann Up mode keeps track of the games you have played, and you can earn progress to a 'rare item'. Then upon receiving that rare item, your card gets leveled up and you can repeat the process as a higher rank and continue to get more and more rare items.

If I can play the update without playing in Mann-Up mode (and without paying), what's the benefit to paying Valve each time?

The $0.99 you spend gets you an item each time. I'm going to say it gives you a hat or misc each time since the FAQ linked above states:

All items unique to Mann Up Mode are cosmetic

So the $0.99 you spend each time gives you a hat/misc each time, as well as adding up to a 'rare-item' though the progress of your Tour of Duty Badge.

However, you do not need to pay the $0.99 each time you play Mann Up Mode,

Tour of Duty Tickets and Squad Surplus Vouchers are ONLY consumed upon successful completion of a Mission. If you do not beat the Mission for any reason, then your Tickets and Vouchers will not be consumed and can be used again later.

So a disconnect or a loss does not consume the $0.99. Therefore you really do get a hat/misc for each ticket you buy.

Personal Experience:

I have played Mann vs. Machine on a regular, non-Mann Up mode, server. I did not receive any item drops upon successful completion of the game.
(Aside, this game mode is very fun and I highly recommend it. Although it can be glitchy, it seems as though Valve hasn't fully tested it when they released it.)

It seems as though STAR has beaten me to sum up what Mann Up Mode is. Watch his vid at 2:35 to 3:15 and 4:24 to 5:28 for some first hand experience as to what Mann-Up mode is.