Dear Kanazawa, do you have a reference for the asymptotic identity given above. I would be quite interested to see how you proved it.

Dear Jarek, many thanks for the nice answer.

Hi Skupers, concerning your approach to Q1, it seems intuitively correct but nevertheless feel shaky with this kind of argument. It seems to me that whenever you remove an open face, you create a "hole" inside $|K'|$ which you can enlarge and push on the sides of the remaining complete simplices of |K'|. But again I feel a bit uncomfortable about the rigour of this kind of argument

Sure, for example take the barycentric subdivision of a 2 simplex (a triangle plus its inside). Then remove the barycenter which is an open 0-simplex. Then this incomplete finite simplicial complex deform retracts to a triangle (a 3-cycle of 1-simplices).

A proof of this result follows from Corollary 9.3.7 of the book "real algebraic geometry" of Bochnak, Coste and Roy.

Ok, I got it. As a special case of your construction, you may take the open ends sine curve which you glue to the x axis along the intersection points.