TE, usually, used when the electric field is parallel to the axes of rods. But thats only just a convention. I use TM for the same thing in my papers.
For the case of the microstrip antenna, you will get the gain you want
On the other hand you can not obtain (simply) the bandstructure of a triangular lattice by using a rectangular unit cell. Because you will loose the rotational symmetry of the triangular lattice. But still it is possible.