This is a DM's call
The only things the rules have to say about monsters using weapons and armor in general is this:
Assume that a creature is proficient with its armor, weapons, and
tools. If you [the DM] swap them out, you [the DM] decide whether the creature is proficient with its new equipment. (MM p. 9)
This and all your other questions boil down to simply asking your DM what they will allow in the absence of explicit rules to adjudicate them. This is not because the DM owns the monster in question, but because it is the DM's job in 5e to adjudicate all the rules at their table.
Shield guardians have hands and are humanoid in shape, so there is probably no issue with a shield guardian using most weapons or most armor. However, there are no rules that say they can, so this is a DM's call. If the DM rules they can wield a weapon in one hand, there should likely not be an issue with them wielding one weapon in each hand.
But, a shield guardian also does not have any weapons in their statblock. So, they cannot be assumed to have proficiency in any of them. The DM must also decide this.
Additionally, as you pointed out, armor would specifically have to be found in the exact right size and shape to allow the shield guardian to wear it which is highly unlikely. Most likely, you would have to have such armor custom made. There are however, no rules for this either, but if we go for the rules for making animal barding we can at least deduce that this would at least be much more expensive than comparable human armor.
But, a shield guardian also does not have any armor or shields in their statblock, So, they cannot be assumed to have proficiency in any of them. The DM must also decide this.
If your DM allows the shield guardian to wear and be proficient in the armor, it should function just like normal armor does on any other creature.
Adventurers League rules
First, a minor correction. The shield guardian is not what counts as the treasure, it is in fact the Master's Amulet that counts as that (with the guardian assumed to be attached to it). Thus there should be no issue with logging anything about your Shield Guardian on the logsheet; you only have to track the amulet. For reference, see the spoiler in footnote 1 in this answer: What is the rarity of an Amulet of the Shield Guardian?
There are no rules or guidance from Adventurers League that give further guidance on this issue (see here for those resources). Thus, it remains a DM decision from beginning to end.
As a D&D Adventurers League Dungeon Master, you are empowered to adjudicate the rules as presented by the official materials (PHB, DMG, MM, etc.). Run the game according to those rules, but you are the final arbiter of any ambiguities that might arise in doing so. (DDAL FAQ v7.1 p.2)
And yes, that might mean that if you move from DM to DM, that each will have a different ruling. That is just the direct consequence of having DM adjudication be strongly present in 5e and moving between multiple DMs.