Try a 56m frame first though. You may fit fine on it if you will accept a more upright position as the saddle to stem drop will be quite low, which suit some well.
My suggestion of a 52 cm frame comes the inseam measurements given and not his height. Please read my psoting and the OP repsones. The OP may have made a mistake on his inseam measurements but as I have not meant him/her I an't pass judgement on that!
The Le mond fit which takes your inseam (77cm) and multiplies it by 0.67 to give your C-T seat tube length.
77*0.67 = 52cm. It is just a starting point. Two customers who are 5ft 8" but have a 31-32" inside leg fitted on a 54cm compact frame and had the right position for them.
It all depend on the effective top tube length plus stem length and the positon you want to take up as too what frame size works best for you. The bar shape you choose make a big difference as well.
I would suspet 172.5mm cranks would be fine and so would 170mm cranks. 175 will also work if you knees are not sensitive. I ride 170 -177.5mm cranks and my knees don't complain.
The inseam given is almost certainly the trouser inseam, so I would not base anything on that measurement. This is a typical error of beginners. I have never met anybody the height of the OP with that short an inseam (it should be measured from the ground right up to the crown jewels) If this is indeed the case, a 52 cm frame would be insane. Likewise, if the OP is indeed a freak of nature and does have very short legs for his height, this will necessarily mean that he has a very long torso and therefore would likely want a longer top tube than would normally be found on a 52 cm frame.
As regards the complaints of sore back and neck, this is generally indicative of lack of suppleness of the rider, so it would normally be indicative of someone who still needs to put in some extra miles before feeling at home on a bike. This is why I suggested and continue to suggest, try a shorter stem, get some miles in and gain some suppleness. Learn from every bike that you own, before trying to change things. 58 cm C/C is NOT that outrageous. I don't think that any knowledgeable person would ever make it their first choice of size, but it is a plausible size for a beginning rider. As a comparison, I have a mate I ride with who is he too 5'11" and he rides a custom made 62 cm C/C frame with 54 cm C/C top tube. Granted he has very long legs, but the bike disappears under him and nobody would ever question the frame size when they see him riding, but when you see the bike leaning up against a building when at a rest while out on a group ride, nobody would guess that it is his frame.
As I said earlier, try the shorter stem. If nothing else, the OP will be able to start to build up his own base of knowledge as to what he likes and does not like.