ㄴ가 이기다

win at something; overcome something

In all eight lines the -ㄴ가 is the tail of 뭔가 'something', not a question ending, so what you are hearing is 뭔가 이기다, 'to beat something'. Often 뭔가 is doing filler work rather than naming a real object, much like English 'sort of' — 오랜만에 뭔가 이겨봤습니다, 'I finally won at something for once'. The compound 이겨내다 in several lines means overcoming a hardship rather than beating an opponent.

Register: casual spoken