Is it common to use 'ceistigh' directly as a verb vs. with 'cuir' or 'agam' (etc.)?
It seems like both are used. Though it's probably even more common to say "I have a question for you", or Tá ceist agam ort.
Because the Irish for the verb "ask" is fiafraigh - d'fhiafraigh na daltaí den mhúinteoir.
But I'm not really sure about the difference between the the English "the students questionned the teacher" and "the students asked the teacher". It's very subtle, perhaps the difference is more obvious if it's "the police questionned the witnesses". Foclóir.ie gives "Chur cheist ar" as an alternative to "fiafraigh" for "ask". Does this mean that "chur cheist ar" is NOT the same as "ceistigh" ?
The difference between the verbs "question" and "ask" really isn't that subtle. If I question your interpretation of this, you clearly can't replace "question" with "ask", and if I question you about your interpretation, there is a clear implication that I am challenging your interpretation, and that implication isn't there if I ask you about it.
Having said that, the NEID includes this example: "'What's that?' he questioned" - 'Céard é sin?' a d'fhiafraigh sé, but the English in that example sound odd to me.
As for chur cheist ar, I interpret as 'I put a question to", which means "I asked" to me, rather than "I questioned".