Master's institution forcing me to deposit $5000 immediately -- what can I do if I accept?

I do not think the quick monetary outlay is common. You seem to be leaning toward acceptance and then backing out if you get a better offer. This may be possible because, in my experience, there is not a centralized system which lets other schools know you have accepted (although perhaps that could be different for Masters programs where you are using federal aid, not sure).

However, I would encourage you to take an alternative approach. Call them and explain that you are waiting to hear from other programs, some of which would be funded. Ask for a deadline extension until the April 15th deadline. In the past I was successful with this strategy, and I know students in our program have also successfully received extensions.

I just want to answer just this part of the question:

c) Let's say I decide to shell out my 5k -- would it be unethical for me to withdraw at a later date if I get into another school? They're not investing anything into me, and they get free money.

if anything, it is (in my opinion) unethical for them to charge this much as a non-refundable deposit. Because of this, if you do spend the deposit I don't think you would be burning any bridges if you rescinded. In fact you just gave them 5k, I don't know how you could dislike someone after that....

a) Increasing common in oversubscribed programs,

b) Very unlikely. I can’t imagine schools sharing admission details. Moreover, if baseball and basketball owners cannot collude, universities cannot either.

c) It is never unethical to accept another offer. The University might not like it but if they want you it’s up to them to make a better offer. Nobody can prevent you from changing your mind, and if a university doesn’t agree with this do you really want to go there?

d) If you choose to walk away, you are burning your bridges with them, not the other way around. See second sentence of c) above.

Although the letter of the April 15th resolution does not apply, it is well worth reading it to understand its spirit.. This passage is particularly relevant:

In those instances in which a student accepts an offer before April 15 and subsequently desires to withdraw that acceptance, the student may submit in writing a resignation of the appointment at any time through April 15. However, an acceptance given or left in force after April 15 commits the student not to accept another offer without first obtaining a written release from the institution to which a commitment has been made.

To me this reinforces my point in c). What is unethical is to keep two offers in play at the same time, i.e. not promptly advising university A that you have accepted the offer of university B.