Hi David
I came across this post somehow, and have just realised that you asked back in February and got no reply. I can’t solve your main achos question for you, but maybe answering the other bit will bump it back up the Forum page so someone else notices it. (Or we could just use the bat-signal to summon @garethrking!)
As for the second bit – I spent a fruitless hour this morning trying to search the forum for my posts when I (a) asked the same question because I was confused by it, too; (b) got a good, helpful reply from @Novem; (c) forgot the good, helpful reply and asked the same question about a year later; and (d) got a good, helpful reply from @Novem (again).
I couldn’t find the wretched things, even with Advanced Search, so here goes:
In your examples, what’s going on is that the original/underlying structure is something like “I went to see what thing it was (that) he was borrowing it.” Beth (“what”) is shortened from Pa beth i.e. “what thing”; and the linking word “that” has disappeared, leaving something that looks much more like the English “what he was borrowing” – apart from the fact that it’s still got that puzzling ei in there.
The other thing that can happen, of course, is that even the ei can disappear, but still leave the mutation behind – so you’ll also see things like Beth wyt ti’n feddwl? (about 4650 results on a Google search) alongside Beth wyt ti’n meddwl? (about 12800 Ghits). The second form is what you’d get if the missing ei were feminine, but it’s also what you’d get if you just didn’t bother mutating in the absence of an obvious reason to do so, which is more likely to be what’s happening. I get the impression that Beth wyt ti’n feddwl? is just a thing that some people actually say, but that spelling it out in full as Beth wyt ti’n ei feddwl? might be a bit fussy/fancy/Literary, but I could be mistaken on that.
More examples in Iestyn’s comment on an old thread here: What am I hearing? - #285 by Iestyn