To answer your first question, yes. The three main forms of questions in order of formality (from formal to informal): "Est-ce un poulet?", "Est-ce que c'est un poulet?", "C'est un poulet?". To answer your second question, no. The article ("un") in front of "poulet" means that we must use c'est.
Chicken in English is also a non countable noun, like pork, beef, etc. If someone is cooking meat, the correct question would be "Is this chicken?" and not "Is this A chicken?", unless the bird is laying in the pan with wings, head and legs... Only "a chicken" was accepted as correct. Hmm...
No, you only use "c'est" with unmodified nouns.
'Est-ce que c'est un poulet?' would work, yes. (I don't know if Duolingo accepts 'est-ce ...?' and 'est-ce que c'est ...?' as interchangeable, but as far as I know they are used pretty interchangeably. There is a SLIGHT difference between 'est-ce que c'es't ...?' and 'est-ce ...?' -- the former is a bit more like 'is it the case that ...?' whereas the latter is more like just 'is it ...?'. However, I don't think 'est-ce que c'est ...?' is as uncommon or weird-sounding in French as its English equivalent is.)
I've never heard JUST 'est-ce qu'un poulet?' though. (That's a bit like saying 'is a chicken?' in English rather than 'is that a chicken?'). The only way I can think of that making sense is if there was more to the sentence after it (specifically, another verb would be needed): for example, 'est-ce qu'un poulet soit aussi important qu'un humain?' ('is a chicken as important as a human?') might work. That's a pretty awkward construction; I'm fairly sure it would still be considered technically correct, though.
Departing from the chicken example, though, 'est-ce que ...?' works perfectly fine in sentences like 'est-ce qu'on ira au centre-ville demain?' ('are we going downtown tomorrow?'). Again, though, it's important that a second verb exists later on in the sentence.
Other than by using 'est-ce que c'est ...?', I'm not aware of any way to work in that second verb in the case of the sentence 'is that a chicken?'.