What exactly is the mechanics of gaining/losing money for protector/protectorat

[Bactron]

Also, if someone is my protectorate, can I later declare war on them if the need arises?

Yes you can still attack them if you want.

[V.T. Marvin]

I personally do not recommend to anybody to try to obtain a protectorate in EB, especially on VH campaign difficulty. Because the protectorate gives almost all its money to protector and because the script instantly gives a huge amount of money to the protectorate the whole thing turns into a indefinite money machine. It will ruin your gameplay experience - being flooded by tens of thousands free mnai each turn is not very funny.
I would recommend to use the force diplomacy mod so that you make your defeated enemy to become your ally and grant you military access (and if you wish even a reasonable tribute) - thus you will get the benefits of a protectorate without being flooded with cash and turning the AI into total passivity.