Yes, übrigens can be used as a stand alone. This is mostly done in spoken German though and there is a somewhat longer pause before the sentence comes. I feel like a simple comma would not do that justice so I'd write
Übrigens... ich habe gestern bei Aldi für eine Dose Tomaten nur 20 Cent bezahlt.
I don't know if it is correct but it is basically just capturing how it sounds. Anyway, as you said, übrigens can also be used as a commenting adverb and as such it can be put into various positions.
[Übrigens] habe ich/Ich habe [ ] gestern [ ] bei Aldi [ ] für eine Dose Tomaten [ ] nur 20 Cent bezahlt.
For questions, the "independent" introduction is by far the most idiomatic way but it can also work somewhere in the middle sometimes
Was willst du übrigens wegen deinem Rattenproblem machen?
This is a bit tricky though and it might sound totally out of place in other questions.
If you want to learn more about the background of the word and related words you can check out this article on my blog.