I'd certainly second "Settlers of Catan" as an example of a game that made me think hard about the interaction of the mechanics and what the implications of omitting any of the rules would be. Also a good game as players are not eliminated, which can help keep the interest levels high.
What's great about Catan is it seems at surface level to be heavily luck-driven from the dice-rolls, but the game mechanics reveal that it's not really about luck, but about your capability of leveraging the good rolls - making the most of them, minimizing the problems from bad rolls, and calculating and planning for the results of future rolls.
Catan is a game of opportunity and risk-planning.
Monopoly is interesting - it is familiar but, as mentioned, most people do not play by the full, official rules.
Not to mention, even with the unofficial rules my family used, Monopoly take three hours or more (can't fit in a single class timeslot), and once you begin to lose, it's hard to recover and you just feel hopeless, as the leaders gain more and more steam.
Settlers of Catan on the other hand, is much more reasonable in length (45 minutes), and I've even speed-played some 4-player matches in under 20 minutes (cramming games between classes :P).
Settlers of Catan also (as you mentioned) doesn't leave losing players in the dust. Most games of Catan I've played, the winner was in 3rd place until the very end, but brought about sudden unexpected victory with several well-planned maneuvers he stored up due to his incredible brilliance and staggeringly handsome complexion. :ph34r:
Seriously, there are very few Catan matches that I remember, where the winner was 1st place throughout the entire game (only the sub-20 minute games). Very few where the winner was first place even 10 minutes before the end.
The Cities and Knights expansion changes that somewhat - the winners are more commonly in the lead earlier on - but Cities and Knights has other great mechanics worth enjoying and comparing and contrasting to vanilla Catan.
Another game of interest is Risk (Lord of the Rings Risk also adds interesting dynamics and is worth playing), but like Monopoly, matches last two hours or so, and would need to be paused and resumed over multiple matches. I do think it's well-suited to being paused and resumed, though, as you can get back into it very quickly and know right where you left of, strategy-wise.
The mechanics are fairly simple though - it's better suited for teaching strategy than mechanics, IMO, but can be used for both.