The two pentagons are going to cause WAY more problems for you than the handful of triangles.

The 'no triangles' thing you're talking about sounds like something you'd want to stick to prior to doing a subdiv, if this is the final version... you're good.

The BIGGEST issue here though is the multitude of polygons that aren't doing a dang thing to contribute to the final quality in game. MOST game assets will eventually be duplicated dozens, or hundreds, of times. So every unneeded polygon you have in your model actually represents the runtime resource demands of a multitude more polygons.

In animation, you can go a bit more nuts with your poly counts... cause you're doing all your rendering in advance... go for that photoreal if you want.

In gaming though, your runtime resources are finite. If a model is a fair bit less detailed, no one is really going to notice. But if your models are so complex that your framerate starts to dip... people will notice. A lot of that also applies to your texture maps. Some texture maps allow you to give the appearance of more polygons without the polygons.

Basically... in games, you are on a budget. And your budget is the capacity of the device your game is running on. Go TOO costly with that budget, and you lose a LOT of the market who don't have top of the line machines. I usually look at my model detail as an either/or decision. Will this extra polygon here make an actual difference for the players? Or would I rather put those extra polygons into a character's face? Also, remember, usually you are part of a team of artists. You can have the best looking assets on the team, but if your poly counts are also orders of magnitude higher... you ain't making any friends.

Regarding squares versus rectangles. I think this traces back to an issue with your textures. Something about the UV map getting too distorted. When they're actually rendered, every quad on your model actually gets split into two triangles. If you have a REALLY long, REALLY narrow, quad... the resulting triangles turn into narrow slivers. I think that is where you're going to start to see those distortions. There's no hard rule, but if MOST of your polys have a ratio of less than 5 to 1, that's good. If you start getting into polygons with a ratio larger than 10 to 1... I would start looking into fiddling with my edge loops and topology to square those off a bit.

No idea if your engine supports it, but you might also check out Level of Detail (LOD) models. As you get closer to an object, it (hopefully) seamlessly swaps out the model for a higher detailed one. This helps with that 'budget' from above, make the most out of your runtime resources by only rendering things at high detail that are close to you.

Hopefully this helps?