Get back the cards I lose in the Card Queen Quest

Triple Triad was designed by Hiroyuki Ito, the battle designer for FFVIII, and wasn't based on a specific card game as far as I know.
It did however spawn a physical version, described here
Also check out Triple Triad Online

It raises stats, though Junctions raise stats far more effectively for the most part. The main purpose of leveling is to raise enemy's tiers--higher tiers (the max tier is actually very low, around level 40 I think?) drop better items. Higher tier enemies also use more dangerous spells or attack patterns however.

For instance, Elnoyle will drop Moon Stones at low tiers, but Energy Crystals (useful for gaining Ultima) at a high tier.

Also, certain encounters have fixed levels. But generally speaking, leveling will actually make enemies harder, though they will drop better item rewards. Much progress in FF8, unlike most other FF games, is related to items directly or indirectly, so they shouldn't be ignored. By fighting higher tier enemies you can more easily synthesize high level magics, weapons or GF ability items.

Grinding in FF8 is rarely direct, aside from AP. (It's also rarely necessary) Grinding to level 100 may seem pointless, but grinding Energy Crystals will give you plenty of Ultima which is both a very useful spell, and the best stat raising junction for several stats.

You can also grind if you WANT to make the game harder. Personally the first thing I do when possible is head straight to the Island Closest to Hell and get to level 100 (which due to FF8's leveling system and the level 100 enemies, does not take long).