Can 得 be used as a stand alone answer? The following question/answer combinations are correct to my knowledge.

他能去吗？能。

他会去吗？会。

他可以去吗？可以。

What about the following question/answer combinations ?

他应该去吗？应该。

他该去吗？该。

他必须去吗？必须。

他得去吗？得。

These examples just sound a bit strange to me. Are they incorrect ? If so then why are they incorrect ?
 And if they are incorrect, how would you answer those questions ? Would you have to repeat the whole statement ? 他应该去吗？他应该去。
It really depends on the context. In literature, all above examples could make perfect sense with the right setting (where the character is from, what situation the speaker finds himself in, even a specific piece of information the author is trying to send, etc). There still are rules but you can really play with it if you’re writing a creative piece.

However, in everyday use, it gets tricky (People of different socioeconomic status talk differently; a language changes in most subtle ways over time). 

The first three examples are perfectly fine, and 


“该 (more natural to the question ‘该不该？’)”, 
“应该。（more natural to the question ‘应不应该？’）” 
“必须（more natural to the question ‘必须吗？【note that ‘必不必须’ isn’t common】’）” 


aren’t necessarily wrong either, though they do sound a bit odd to my ear. 

I would use “（他）应该去。/（他）必须去。/（他）该去。” instead. The subject could hide itself but the verb should not, in this case. But again, this is not a final conclusion, and you may find yourself in different situations where the usage of all three of them is acceptable.

As to the ultra tricky “得”，I also find “（他）得去” more natural. However, because “得” feels so northern, I have no problem picturing an old-fashioned northerner using the word alone as an answer, when the speaker really wants to stress it.

I hope this is helpful, and I strongly suggest you look up words/phrases in dictionaries or classic literature (or a reliable grammar book, the state news broadcasting, things like that). The Internet while providing good examples of slangs could be misleading as it’s filled with grammar mistakes. And please do not trust all tv shows...