(i) Your sense of TE and TM seems correct to me. In an experiment, the angle of incidence is set by the orientation of the horn relative to the FSS. In many anechoic chambers, there is a rotatable platform, sometimes robotic, that slowly turns the FSS as the frequency response is being measured for each angle. Usually standard-gain horn antennas are used that transmit a mostly linearly polarized beam. To change from TE to TM, simply rotate the antenna 90 degrees. Be sure your illluminating horn is sufficiently far from the FSS to get a nearly plane wave wavefront at the FSS.
(ii) I don't know CST, but I still think I can answer this question. CST probably only provides options for TE and TM because you can make any polarization from just a combination of these two. For example, if you want a linearly polarized beam at a 45 degree angle between the TE and TM, you would simply set the amplitude of both TE and TM to 1/sqrt(2). If you want a circularly polarized beam, you would do the same, but give the two a 90 degree phase shift between them. If you want to see a bit more on polarization, TE, and TM, see Lecture 3, Slides 9-21 here:
**broken link removed**
(iii) HFSS does have this option. I think mostly any tool will have this option, including CST. I think CST is very expensive compared to HFSS and in my experience HFSS produces results that more closely resemble what we measure in the lab. CST may be better for very large structures, but HFSS now has a time-domain solver. For periodic FSSs, it will be hard to beat HFSS.
(iii) Yes! See above.
(iv) I think you can vary the polarization angle. Check to see if you can vary the amplitudes of both TE and TM. If not, but I think you can, do it manually. Figure out the two amplitudes, simulate the TE, simulate the TM, and then add the results.