DM Rulings and 1e AD&D Rules Combine for a reasonable outcome
This answer requires a caveat: there is no strict rule about critical hits in the 1e books, to include Unearthed Arcana, but a lot of 1e DM's use critical hits of varying sorts. The rules in the DMG, p. 61, explicitly put critical hits outside of the rules.
1: Can a Frost Brand weapon freeze someone solid?
While that isn't in the item description (p. 165, 1e DMG), your DM can to rule that it can, based on the rule of cool (sorry for the pun). As you were dealing with a Frost Giant deity, he may be able to create that effect with one of his powers, beyond the sword's basic effects.
- If so can sunlight melt the ice that's been formed from the weapon?
Sure, sun melting ice isn't against the general primary world rules of 1e AD&D. How long it takes is up to the DM.
2: If the character is frozen, would his Ring of Regeneration not be able to regrow the top half of his body to the bottom half?
Per the item description (p. 130 1e DMG) the standard ring can bring you back from death and restore limbs. Knitting you back together is well within that capability.
If it is a vampiric regeneration ring, it does not perform the knitting function, but instead acts as a ring of life steal on each successful melee attack. (You get some of the damage you do back to your HP total). Unless the DM specified that it was of the vampiric sort, then the ring ought to eventually knit you back together.
3: Are system shock checks really necessary when someone has a regeneration ring on?
Yes, if the DM is using a critical hit table that calls for system shocks, or if his judgment on your body being chopped nearly in two calls for it. I agree with him, but, it isn't strict RAW for two reasons:
System shock is for surviving certain effects, (p. 12 1e PHB)
which do not include taking damage short of death.
Critical hits aren't in the rules as written.
That said, if you look at the trauma of a petrifaction or polymorph spell on a body, which does call for system shock, his ruling that being nearly cut in half requires a system shock check is well within reason.
1e is very much a "rulings over rules" system. This is supported in numerous places in the DMG.
Even if they fail the check, won't they just regenerate X amount of hp and then be able to return to the fight?
Yes, eventually, but System Shock is its own mechanic. The DM may require more time for the regeneration/mending to take effect in the case of a critical hit which, again, is completely within his discretion since its his critical system at work.
RAW note on Critical Hits
Critical hits are NOT supported by the original rules (DMG p. 61, Combat section, second paragraph). That said, there is a lot of homebrew in 1e games. There was an article in Dragon Magazine #39 that covered a table of critical hits and critical fumbles. A lot of DM's I played with in the 70's and 80's included critical hits, even though Gary Gygax' rule book specifically addressed them and did not include them.
Bottom line: your DM's ruling is legit and reasonable. Where he got the "it froze you solid" is unclear ... though it's a great application of "the rule of cool" and displays a fine narrative style.