Real-world-damage is classified in several different ways in D&D 3.5

Hit point damage

What are hit points? Hit points are your capability to not take the fatal blow, but are also how wounded you are.
Are these two definitions a dycothomy? Well, as with every abstraction I'm prone say yes.

Poisons that only get applied by harming you enemy and drawing blood need HP damage to be dealt. At the same time most other mechanics work really well by comparing HP with how much stamina you still have - when you have none, you can't parry properly and you're k.o.

Cure spells and potions do just what natural healing does, just faster: it restores HP. (Maybe with less scars, maybe with more, who knows?)

Ability damage

This recovers really slowly over days of complete rest (or with specific magic) and I'd go with this to represent having a limb broken, except it isn't specific to a location.

Limb severing

While the game states no in-combat way to have a limb severed, several spells consider the opportunity. Polymorph efefcts have severed limbs take their former form or not depending on which kind of spell or ability you have been using and regeneration (a 9th level spell) is the only way to restore a lost limb save a miracle or some shenanigans involving different spells.
Losing a limb is pretty definitive and shouldn't happen during a combat.

So, breaking an arm does not look like an option.

What would have been the rules-friendly way to handle this? Well, not ruling a broken arm in the first place, and settling for a more temporary form of nonlethal damage, like straining a muscle. Nothing in the grapple rules allows you to disable your opponent that badly. Again, this is a problem that lies in how D&D abstracts this specific thing, and D&D is all about "no consequences until you get to 0 hp and then the only consequence is that you stagger/faint or what have you (a limited list of outcomes that does not match with reality).

Since the D&D rules work like a physics simulator that uses its own unfamiliar phisics, it sure breaks our expectations, but IMO it's better to break immersion where the rules say than having to face unstated consequences like having to decide how inconvenient it is to lose a brawl after it has happened, because the unpredictability of consequences destroys the risk assessment capabilities of the players and makes them unable to make informed decisions.

The use of the various types of tools (including the Gaming Set) do not rely on a particular skill but instead simply use an ability score appropriate to what you are trying to achieve.

From the first paragraph on p. 154 of the PHB:

For Example, the DM might ask you to make a Dexterity check to carve a fine detail with your woodcarver's tools, or a Strength check to make something out of particularly hard wood.

Having proficiency with a particular set of tools allows you to add your proficiency bonus to the roll in addition to the bonus (or penalty) granted by the ability score that is being used.

As for how the situation would play out, p. 174 of the PHB describes how to handle a contest between two individuals:

Both participants in a contest make ability checks appropriate to their efforts. They apply all appropriate bonuses and penalties, but instead of comparing the total to a DC, they compare the totals of their two checks. The participant with the higher check total wins the contest. That character or monster either succeeds at the action or prevents the other one from succeeding.

If the contest results in a tie, the situation remains the same as it was before the contest.

In your particular case it would depend in part on what game they are playing and what sort of approach they are taking. They could maybe use Charisma to try and bluff their opponent. Or they could try and read their opponent with an Intelligence or Wisdom check. Maybe even Dexterity to try and discreetly slip a winning card into their hand.

Participants do not have to use the same ability check: the NPC could try to bluff the PC with a Charisma check while the PC tries to read the NPC with Intelligence.