Yes, that's rust.

We've had a ton of cool wet weather so is this something I should be concerned about or will it naturally get better as it warms up?

I'm already practicing the normal things for a healthy lawn, and compared to neighbors/friends it's the thickest best lawn ever, so I don't want to see it get destroyed by rust!

Put some fertilizer on those spots and the grass will grow out of it. Starter fertilizer works best for those situations.