Thank you for the answer. 1. I need a bit of help with the answer for the second question: "The triangulated surface in $\mathbb{R}^3$ with one boundary component which is not a disc". Since the complexes I'm looking at are compact, I guess this would look something like a two-dimensional doghnut in $\mathbb{R}^3$. Then doesn't the boundary consist of the inner/outer circles and is not connected? 2. I want to try to modify the second question to make it work, so I hope you won't mind me not accepting the answer yet.

Thank you for the comments. The answer for both questions is a yes, and we really don't know how the intersections look like yet.

Thank you for the answer. Too bad it is hard to play around with algebraic matroids...