1)
食べてきた=ate (it) and came back home
食べた=ate (it) ※Tomorrow, he might be going to eat Japanese sushi at the same hotel where he and his friend are staying now.
2)
今年の初めてのカラオケで、楽しかった。
I think今年初めてのカラオケで、楽しかった。is natural.
それは今年初めてのカラオケだった(←であった)。そしてそれは楽しかった。=It was the first karaoke this year. And it was fun. →今年初めてのカラオケで(あって)、楽しかった。It was the first karaoke, and it was fun. ※で is not a particle. It is a conjunctive form of auxiliary verb だ.
3)
なんとなく=somehow / vaguely / faintly
週末は全然勉強していなかったから、なんとなく不安を感じた。=Since I was not studying at all during the weekend, I felt uneasiness vaguely. / Because I didn't study at all on the weekend, I felt vague uneasiness.