こんな状況で、俺は勉学に励んでなんかいられるのか？
can it be that i (do such a thing like) focus on my studies?
Is there a non-contracted form of this phrase?
It could be 勉学に励んで など いられる もの な の か if this could be of any help.
I take this か as self-questioning. The words sound like the speaker is worrying about his own (near) future.
The なんか adds a nuance to treat the thing mentioned right before it lightly. In this case, the speaker is degrading the activity of studying because other things he's facing right now are more important or at least taking up too much time and occupying his mind too much.
いられる is continued from 励んで. It's a form made from 勉学に励む, 勉学に励んでいる (state/progression), and into 勉学に励んでいられる (potential/ability)
[Reply to the additional question]
is that the case when it doesn't end in a negative statement? Compared to "adds emphasis against an assumption or statement by others and is used only in negative statements" from なんか after て- form of a verb
I see that the point of the other thread is also degrading. I find the degrading なんか is not limited to negative or questioning sentence: うちの子なんか寝てばっかりです. なんか could assume a proud tone while appears to be degrading: (I can't really come up with a good example, but for the time being..something like) うちなんかまだこれからです. Also it's good to keep in mind that なんか is more informal than なんて.
なんか
２ ある事物を例示し、それを軽んじていう意を表す。…など。「彼の言うことなんか聞くな」「君になんかわからない」（デジタル大辞泉 | なんか）
なんて
１ ある事物を例示して、それを軽んじたり、婉曲 (えんきょく) に言ったりする意を表す。なんか。…などということは。「手伝いなんてできるか」「本気にするなんてばかね」（デジタル大辞泉 | なんて）
