Hey I will try to answer a few if I can. This is a really obscure subject and even the big textbooks from Friedman don't cover it in much depth. Unfortunately I'm moving this week and all my books are packed away in boxes, so some of this is off the top of my head. I'll try to reconfirm later on and maybe provide page number references etc.
- I assume that any of 51/52/57/63 can control both 40mm and 3"/50 RF, is that correct?
As I understand it, the Mark 52 was a stopgap designed for small units and the 3"/50 (non RF). The Mark 51 was a really basic director that could be adapted to control everything up to 5" guns. The Mark 57 was designed to be a radar-assisted version of the Mark 51, with four Mark 57 on BBs and CBs to help defend against saturation attacks. Mark 57 could control the 5" guns in a pinch, and I think it was modified postwar to control 3"/50 RFs. Mark 63 was only used to control 40mm Bofors quads (the twins were too small to mount the radar component) and the 3"/50 RF postwar.
- Is there any difference in capability between using Mk57 and Mk63 to control the Bofors/Mk33/34 gun? I mean it looks like their difference is just 57 installed the radar on the director and 63 installed the radar on the gun mount.
From what I've read, Mark 63 wasn't particularly well liked by crews and the installations didn't last long postwar. The Mark 57 was a superior system. The radar marks on Mark 57 and Mark 63 were different (Mark 57 used radar Mark 34 and Mark 63 used radar Mark 28). I'm not sure of the performance differences between the radars, but I bet a lot of the issues encountered with Mark 63 were due to the separation between the director (calculating lead, height, and azimuth) and the radar (calculating range) on the mount. On-mount ranging radar lasted into the cold war with the GUNAR system on the 5"/54, but I'm not an expert on those systems (maybe a question for erik_t). Either way it doesn't seem to be a thing anymore so that probably answers that question.
- Does the US Navy export the 57/63 during the late-40s/early-50s? Or just 51/52?
It wouldn't surprise me if the Mark 57 and Mark 63 systems were removed from export vessels before transfer, but I have no evidence to confirm one way or another.
- When did AN/SPG-34 replace the old radar? And do they use it with the Mk63?
I am not sure about this but I bet Friedman's Naval Radar has the answer (my copy is packed away).
and another question that is not about the director, when did the enclosed Mk33 become available?
If you mean the Mark 33 gun director system, the first of those directors were aboard the FARRAGUT class but they were the open-topped version which seems to be externally identical to the Mark 28 aboard the early cruisers. I'm not sure which class first received the enclosed Mark 33. It might be the GRIDLEY class but I'm just not sure at this time. If I can find out I'll update this post.