Learn Russian: Это очень грубый ответ.

Here, это functions as a standalone subject meaning "that is" or "this is". When used this way, it always stays in the neuter form это, regardless of the gender of the noun that follows. If you wanted to say "this answer" as a single phrase, you would use the masculine form: этот ответ.

How would the meaning change if I rearranged the sentence to Этот ответ очень грубый?

The focus shifts slightly. Это очень грубый ответ means "That is a very rude answer." By contrast, Этот ответ очень грубый means "This answer is very rude." In the first, you are identifying what the statement is; in the second, you are pointing out a specific answer and describing it.

Could I use the adverb form грубо (rudely) here instead?

Not in this exact sentence, because you are describing a noun (ответ), which requires the adjective грубый. You would use the adverb грубо if you were modifying an action (a verb). For example, if you said "He answered very rudely", it would be Он ответил очень грубо.

Can грубый describe physical objects, or is it only for behavior?