hi
about ur question 1
i think u dont undrstand how work eigenmod
i send u aexample of rectangular cavity from balanis electromagnetic book
and simulation of this example
solution in book
resonant frequncy=9.28Ghz & Q=7758
solution in HFSS
resonant frequncy=9.2707Ghz & Q=7777.9
this example can help u undrstanding eigenmod
note that the real part of frequency in eigenmod is resonant frequency and image part depend loss of matrial and determinant Q factor
i simulate a simple cylinder R=4cm & h=2cm and assume ur matrial is pec(perfect electrical conductive)
resonant frequncy(firstmode)=2.8642GHz and Q is not available becuse material is pec (without loss)
i send u 2 simulation and image of example in balanis in a zip file
excuse me my english is not good