It's assumed to be multiplication of 'function of only x' and 'function of only y' to solve the differential equation. The method is known as 'separation of variables'
Circular waveguide video has been uploaded. General solution in cylindrical coordinate system ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-ofScinEG9U8.html and for circular waveguide ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-PHNjH653OcQ.html and parallel plate waveguide is also available Thank you
TE is transverse electric, means Electric field is only in transverse I. e. Ex and Ey but Ez is zero. hx and hy are transverse components of Magnetic field.
TE10 is not dominant mode it's TE01, if a=1cm, b=2cm then, TE01 mode: m=0, n=1 so, cutt-off wavelength=2*b=4cm while, TE10 mode: m=1, n=0 so, cutt-off wavelength=2*a=2cm Thus TE01 is the dominant mode.