"Tá go maith!" translates as "It is good" and means "Thats fine" in English, correct? Why isn't it "Tá sé go maith!"? Is it common to drop the subject like this in Gaelic? Is "Tá go math!" idiomatic?

Does "Tá sé sin go breá" have the same meaning? Can I say "Tá go breá" or "Tá sin go breá"? If so, do they have the same meaning?

Tá go maith!
means 'Good!' or 'Grand!' in a general sense is indefinite without specifying anything in particular.
Tá sé go maith or Tá sé sin go maith is definite and means 'It is good'
If you want to say more than just 'grand' or 'good' and wish to convey 'beautiful' or 'great'
then put in 'breá' where the 'maith' is.
Tá go breá! - indefinite as outlined above
Tá sin go breá. or Tá sé sin go breá. - definite as outlined above

Although 'Tá go maith' you might say every day, for the other one I would usually go for 'Togha!' or 'Thar cionn!' for the indefinite - that everything is fantastic, etc.
and
'Tá sin go breá' for the definite - referring to how wonderful, fantastic whatever is said or whatever object we are talking about at the time.

Hope you can make sense of that for starters... but others will shortly help out and clarify anything that needs changing, etc.

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Bí cinnte de go nglacfaidh triúr le gach aistriúchán a thabharfar.
Be sure to get three in agreement with a translation given.

To kind of summarise what Braoin has said, the versions without a subject correspond to interjections in English:

Tá go maith ! "OK!" or "Alright!"
Maith go leor "OK!" or "Alright!" or "Good-o" or "Righty-ho".
Ceart go leor. "OK!" or "Alright!" or "Right-o" or "Righty-ho".
Dona go leor. "Oh, no", "What a pity!" or "What a shame"
Togha ! "Great!" (where the Irish say "Grand!")
Thar barr ! "Excellent!"
Thar cionn ! "Outstanding!"

In contrast to these, the versions with sé are your normal (unmarked) sentences.

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Múinteoir Gaeilge - Irish Teacher
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

SO BE IT! seo míniú eile atá le Tá go maith!
often used when someone does something against the advice of another

Yes, I don't think any language is free from the use of irony.

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Múinteoir Gaeilge - Irish Teacher
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).