The use of 'which' is completely throwing me off. I am not confused b/w the use the 'which' and that of 'that', but rather, I do not understand what 'which' can refer to vs. what 'which' cannot refer to.

Does 'which' always only refer to the noun immediately preceding it? Or can it refer to a sentence or clause as well?

Eg:

1) Our understanding was that she would continue to cover the costs of school enrollment and room and board for the children, which she verbally agreed to pay with the rest of the family.

2) Our understanding was that she would continue to cover the costs of school enrollment and room and board for the children, both of which she verbally agreed to pay with the rest of the family.

Does 'which' refer to 'children' in 1 making it wrong?

I am so confused! Thanks for your help.

"Which" can be a little tricky. The rule is that it has to refer to the closest preceding primary noun. It's often the case that this is the immediately preceding noun, but not always.

So, when is the primary noun not the immediately preceding noun? When some description needs to be attached to that primary noun to explain it adequately.

She agreed to cover the cost of the tickets, which was very high. -- technically referring to cost b/c "of the tickets" is a description of the type of cost
It would be cleaner, though, to say "She agreed to cover the tickets' cost, which was very high."

Sometimes, we can't easily move the descriptor noun earlier as a possessive (as we did above) and we have to keep it as a prepositional phrase. eg:

After falling down in the middle of the runway, she maintained her aura of detachment, which inured her to the laughter. We wouldn't say she kept her detachment's aura - we have to use the prepositional phrase there, but the "which" clause really modifies aura, not detachment.

You'd be unlikely to see something like your sentences on the official test b/c it's a bit too complex. Most of the time (on the test), if this issue comes up, it will be presented as a single prepositional phrase following the primary noun.

Can we use the above mentioned logic for 'that'? Please explain. Thanks in advance.

Yes, you can - "that" can also indicate the beginning of a noun modifier and follows the same rules: it should refer to the closest preceding primary noun, which is often the closest word, but not always.

Remember, though, that "which" and "that" are not interchangeable...