Learn Hardcore Korean: mateueseo gyeranirang dubureul sasseo.

에서 marks the place where an action happens: 마트에서 = at the mart / in the supermarket (the buying happened there).
에 usually marks a destination or a location of existence:

마트에 갔어 = I went to the mart.
마트에 있어 = It’s at the mart.

Is the 이 in 계란이랑 the subject marker?

No. In 계란이랑, the 이 is part of the connector 이랑 (used after a consonant). It means and / with.
So 계란이랑 = eggs and / eggs with (as a list item), not eggs (subject).

Why is it 계란이랑 두부를—why does only 두부 have 를?

In Korean, when you list multiple objects, it’s common to put the object particle (을/를) only on the last item.
So 계란이랑 두부를 샀어 = (I) bought eggs and tofu.
You can mark both, but it can sound heavier/emphatic: 계란을 두부를 샀어 (less natural as a simple list).

Could I drop 를 and just say 계란이랑 두부 샀어?

Yes, very often in casual speech the object particle is omitted if the meaning is clear:

마트에서 계란이랑 두부 샀어.
That said, keeping 를 is also very normal and can sound a bit more complete.