Is it possible for me to put the sound peg back into my violin properly myself?

You don’t know a luthier who can do it on the cheap or for barter?

I will gladly barter… but I know no luthier

Depending on whether it’s just slipping or whether it is so loose it’s rattling, you may just need to put some resin on the peg, recreating the friction required to hold the peg – this is something I have done with success.

Also, it is now possible to get geared violin pegs – that look almost exactly like normal violin pegs, but work on a geared system like a guitar peg – I found you a video on them here

Oh- the tuning pegs are not at issue. (But that rosin tip will come in VERY handy in the future!)

Here is a cut out view. Evidently I erred in calling it a sound peg- it is a sound post.

There seems to be a special tool for this- and placement is vital for proper sound. :/

Ah, yes, different matter altogether… I’m afraid I would definitely recommend a specialist for that, sorry! Your “local” music shop is likely to have a connection with someone that may be able to help – that would be my first port of call. Another idea would be to take it to someone who knows something about wood? They will have a respect for the instrument and likely to know something about the best place structurally to put it?