I did not see the comment section under Yaakov Baruch's answer to this question. I think these ideas are interesting enough to be posted as an answer; I'm making it comminuty wiki. This provides a negative answer to your first question: all cells have to be triangles or quadrilaterals.

I call “block” a finite union of cells.

Consider a cell with at least 5 sides. It defines some triangular blocks based on each side, that I will call “ears”; see Yaakov Baruch's answer for pictures. Note that these ears can be degenerate: even if no two sides are parallel, they can be infinite, or, in a different mindset, they can contain the cell itself. This does not affect the proof, provided we consider by convention the area of such degenerate ears to be larger than that of the cell. The following fact is a rewriting of Yaakov Baruch's points a) and b).

Fact (Baruch, 2011). For any cell with at least 5 sides, at least one of the ears has area less than the cell itself.

I must admit I do not fully understand the proof of the pentagonal case (why would the wiggling of the line increase at least one of the changing areas?), but it is given in a comment to their answer, and the induction step is described in the answer itself, with some pictures. The fact implies directly that there is no cell with at least 5 sides.