Where can I find a proof of the fact that a finitely generated group is residually finite if and only if it acts faithfully on a locally finite rooted tree?
One way is actually quite easy - that every group acting faithfully on a locally finite rooted tree is residually finite - and can be found in a paper of Mark Sapir.
He then implies that the converse is a result of Kaluzhnin, and gives a reference as,
However, I have more than just skimmed this paper (it is 76 pages long, and I couldn't find a searchable version) but I couldn't find the proof of this result.
(As a side note, the paper of Sapir is actually very interesting, and well worth a read. Although I found his proper paper of the result easier to follow...)