Hello! New homeowner here, excited to start tending my own yard!

I checked the FAQs and did some searches to gather some info, but really I'm in the bushes (pun intended) when it comes to lawn care at this point in time. I had some questions on some things in my new lawn I was hoping some of the more experienced veterans could give me some pointers on!

I live in denver, CO, not sure what type of grass we have or what's been done to it, previous owners had dogs in back yard.

Some dry spots, does it just need extra water loving? More seed?
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Part of a large playground set was left on lawn for a few days, flattened grass, it's now a lighter color and more moist, what is the best TLC that can be done for this area?
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On the sides there is longer grass with fluffy tops, is this grass or weed, do I mow it like normal or treat it with anything?
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Is this a weed or part of a tree or plant?
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Is this dying/dead or is it normally late to green?
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My 3 year old likes this 'plant', but it's a weed right? Before I pull it...
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