not sure what heavy means. our gaming sessions are around 40-60mins, sometimes longer.
Neither of us are boardgamers - though I love TCG like MTG and the ilk. My gf hasn't really played board/card games before. We play League of Legends together - and I'm thinking of branching out and picking up a board game/card game.
We are competitive, but we also like co-op games. we are quite cerebral and like complexity. i like more interaction than not
right now im eyeing pandemic and summoner wars
To me, heavier typically means more complex rule sets and generally more taxing on the brain. For instance, I consider both pandemic and summoner wars somewhere around light to medium on the scale.
If you think your gf will like a card game like mtg, then I would highly suggest netrunner.
The coop games I suggested are basically big puzzles that you can work as a team to resolve, but may be too light/easy for your tastes, since the difficulty in them basically falls upon random chance.
Innovation is a great choice since it is highly interactive.
1955 is a great two player game with simple rules but a lot of thought goes behind each move. Even more of the same can be said about carcassone and I will second it as a great recommendation.
Jab is a different take on your usual card game and you'll enjoy it if you like playing stuff like speed/ers/slapjack with a normal playing deck of cards.
Summoner wars ends up kinda like chess, but each side gets their own specialized pieces they can play with. Still a great choice as well.
I'd also recommend finding a video review/playthrough of a game from dice tower or bgg to get a feel of how any particular game is played.