So,

You are beautiful == Tá tú álainn
You are a beautiful girl == Is cailín álainn tú
I think you are beautiful == Ceapaim go bhfuil tú álainn
I think you are a beautiful girl == Ceapaim ????

Not sure how to finish the last one. What is the proper structure for "Ceapaim <copular phrase>"?

gur

Ceapaim gur cailín álainn thú.

So, for tá you would use go bhfuil and for all other verbs (or most other verbs) you use gur.
Is that correct?

Níl.

You use go (+ eclipsis) for all verbs in all tenses (except past tense)
You use gur (+ lenition) for all verbs in past tense (except a few irreg. verbs [bí, téigh, feic, déan, faigh, abair] which use go in past tense, too.)
You use gur(b) (w/o lenition) in present tense copula clauses, gur(bh) (+ lenition) in past tense copula clauses.

In the case of gur(b) and gur(bh) I assume the extra letters are for when the next word begins with a vowel??
So, in the above example

Ceapaim gur cailín álainn thú == "I think you are a beautiful girl"
Ceapaim gur chailín álainn thú == "I think you were a beautiful girl"

Also,
Cepaim gur ríomhairí tábhachtach == "I think computers are important" or "I think computers were important".

In the last case, since "r" does not take lenition the only way to tell if it's supposed to be past or present tense is context.

Do I have this correct?

No. In this case you have an adjective (important) as a predicate so you need the verb bí, not the copula.

Ceapaim go bhfuil ríomhairí tábhachtach – ‘I think computers are important’
Ceapaim go raibh ríomhairí tábhachtach – ‘I think computers were important’

Perhaps, to put emphasis or to mark it is a subjective judgment, you could use a structure like in is deas é sin – but I think you need a definite noun then (not sure though, can indefinite noun be a subject of such construction?), so something like:
Ceapaim gur tábhachtach [iad] na ríomhairí – ‘I think the computers are important’ (as alternative to ceapaim go bhfuil na ríomhairí tábhachtach)

Cf. The copula and adjectives on GnaG.

DOH!!!
That's right, my computer example is a bi sentence, no a copula. Slipped by me.