The standard form is "tá" (e.g. tá mé, tá tú, etc).
You'll never see "bhfuil" on its own. It'll always be "go bhfuil", which roughly translates "that is" (e.g. "Deir Jack go bhfuil sé ag teacht" / "Jack says that he is coming").
"Atá" is a little harder to explain. It's used when it isn't beginning a sentence or a point (e.g. "céard atá...", "conas atá...", "sin an rud atá ag cur isteach orm".
"Atá" is a contraction of "a tá," with the "a" introducting a dependent clause. You'll also see "a bhfuil," depending on the content of the sentence. There's a good explanation here: http://www.irishgaelictranslator.com/translation/topic16980.html
It could not be "cén atá an bhear".
You could say "cén rud atá ag an bhfear?" - "what thing (is it) that the man has?"
"cén" has to have a noun, it doesn't stand alone (therefore "cén rud"). You need both the verb "bí" and the preposition "ag" to translate "have" (therefore "atá ag").
"an bhear" is just wrong - there is no feminine noun "bear" that would become "an bhear", and the two possible nouns with fadas ("béar" - "bear", as in "polar bear", and "beár" - "bar" as in "pub") are both masculine, so they would not be lenited after "an", and if it's just a typo for "an bhfear", there is nothing in "cén atá an bhear" to eclipse "an fear" - it's a preposition like "ag" that eclipses a definite noun.
You'll hear agam pronounced am in the middle of a sentence, as in "Tá aithne agam ar Charolette" which is "I know Charolette" but what you'll hear is "Taw ay-na am air Charolette". Language is spoken and writing came later. I've not seen this with agam at the end of a sentence. "Tá mé carr agam" - "I have a car" which will sound like it's written.
That's just how Duolingo works - it's trial and error.
With regard to your second question, I would imagine saying that the insect is beside / in front of / behind the man before I'd say that it is at the man, but if you wanted specifically to say it is "at", the man then yes, you would say it in the same way as that the man "has" the insect.
The relative particle 'a' should be used together with a verb, e.g. "Cé a itheann iasc?" which translates as "Who eats fish?", or more literally, "Whom is [it] that eats fish?" However, the verb form 'tá' is combined together the particle, creating 'atá', e.g. "Cé atá ag an doras?", literally "Who is [it] that is at the door?" I think 'táim' and 'táimid' are also joined: 'atáim' and 'atáimid', respectively. With almost all other verbs, the particle and the verb are two separate words.
I have to confess something here...
I’ve read all of the comments talking about the technical components of the sentence.
Meanwhile, I was just proud that I could figure out that the “ag an bhfear” was “at the man”, and could actually get one right for a change! :)
Small victories need to be celebrated, too, ya know!