This could be thought of in terms of the impulse response of the wall.
If the impulse response was non oscillatory (say, a decaying exponential) then the sound on the other side would be a low pass filtered version of the impinging wave--whether it is an impulse or something else.
Otherwise the wave on the other side could include the resonant frequencies--which would be summed together with the low pass filtered impinging frequencies (assuming the impinging wave contains the some of the resonant frequencies ).
1) Does a sound wave always (no matter what frequency) needs of the
interaction (set in motion particles) with the element the wave is
Yes--in all cases this would depend on vibrating particles since sound is a mechanical wave.
2) Is the sound that gets transmitted to the other side of the wall
only a result of the vibration transmitted through the wall, or also a
combination with the sound that is radiated by the wall at its
resonant frequencies after having been excited by the impinging sound
(As you say) It is a combination of the sound that is radiated by the wall with the wall's resonant frequencies after having been excited by the impinging sound wave? So the sound on the far side of the wall would consist of the frequency components of impinging sound wave with their amplitudes modified by the frequency response of the wall including any resonant frequencies.