Given my understanding, it seems like "どちら" has the exclusive meaning of "which one (of two)" and "どれ" means "which one (of many)". Does どっち include both of these meanings?
Also, do all of these strictly mean which one instead of which ones? Is there a different word to use when you want to say which ones?
「どっち」 can only mean "which one of the two"; No exceptions. It is simply the colloquial way of saying 「どちら」.
「どちら」, however, has another meaning of "which place" or "where" out of many places. When it is used for that meaning, it is NOT interchangeable with 「どっち」. Instead, it becomes interchangeable with 「どこ」.
To answer your second question, which is actually more complicated than you seem to think because of the place-related meaning of 「どちら」, I am going to say that GENERALLY, 「どちら」 and 「どっち」 strictly mean "which one", not "which ones", but when 「どちら」 refers to a place, it can mean "which places" in the plural. 「どれ」 can swing both ways between singular and plural.
Hand-made example sentences:
You are asking someone which one of the two candies s/he wants.
A: 「どっち/どちらのお[菓子]{かし}がほしい？」
B: 「こっち (or そっち)。」 B will say either of the two by pointing a finger to one of the two candies.
A and B are in a clothing store selecting tee-shirts to buy.
A: 「どれにする？」= "Which one(s) are you getting?"
B: 「この[青]{あお}いのと、あの[赤]{あか}いのにする。」= "This blue one and that red one."
B could have just chosen one instead. This is what I meant by "どれ can swing both ways".
B is going to Europe.
A: 「[今回]{こんかい}はどちらを[廻]{まわ}られるんですか。」= "What places/countries are you going to this time?"
B: 「フランス、イタリア、スペインです。」
