Quiero beber tres vasos de agua.

Verbs of desire like querer are normally followed by an infinitive when you talk about wanting to do something yourself.

• Quiero beber = “I want to drink.”
  If you used the subjunctive (Quiero que beba), you’d be expressing “I want (someone else) to drink.” So to say “I want to drink,” always use querer + infinitive.

Yes. Both mean “to drink,” but:

• In Spain, tomar is more common in everyday speech: Quiero tomar tres vasos de agua.
• Beber is perfectly correct but can sound a bit more formal or technical.
  The meaning remains the same.

Agua is an uncountable noun, so you can’t directly count “waters.” Instead, Spanish uses a container or measure + de + substance:

• un vaso de agua = one glass of water
• tres vasos de agua = three glasses of water
  If you just want some water without specifying amount, you say agua on its own.

Yes. Spanish often adds a reflexive pronoun to verbs of consumption for emphasis:

• Quiero beberme tres vasos de agua.
  This doesn’t change the core meaning (“I want to drink three glasses of water”) but adds a nuance of “drink it all up” or “finish them.”

When you have an infinitive after another verb, the pronoun attaches to the infinitive. Here, tres vasos de agua is masculine plural, so:

• Quiero beberlos.
  That single word beberlos = “to drink them.”

No. With measurements or containers you always need de between the container and the thing measured:

• tres vasos de agua, una taza de café, un litro de leche.
  That de is essential to mark the relationship “container/measure of substance.”