A wobbly mast certainly would cause variation in FreeSat reception - the satellite is rather a long way away (about 36000 kilometres, a tenth of the way to the Moon). As would any imperfections in the original alignment.

BBC4 changed channel on Astra about 8 weeks ago- have you tried rescanning for it?

The box is unlikely to be damaged by misalignment, but its original quality may be suspect.

I don't know why RTE demand subscription for FreeSat channels when the BBC don't.