New homeowner completely clueless to lawncare. Need to seed now and having hard time finding some answers

I'm a new homeowner and it's my first having to care for a lawn. I've tried doing some research and I see that now isn't the best time to plant seed, but there's a bunch of small dirt patches through the yard that get muddy and my dog always finds, so I need to take care of them. There's a few basketball size dirt patches and a bunch of smaller random patches mixed with some grass.

I'm in PA, suburbs near Philadelphia. I don't have an irrigation system, just a sprinkler, which can't reach all the spots at the back of my yard. From what I can tell, my best option is Turf Type Tall Fescue mixed with Kentucky Bluegrass. I have a few questions though:

• Do people just buy two bags and mix it themselves? My parents gave me their old seed spreader, so I just buy the two bags, dump a bit from each into the spreader and have at it?
• It seems like planting seed now will mean there's a good chance it won't live through the whole summer. Does that mean it makes more sense just to get some cheap seed from Home Depot/Tractor Supply and then buy the nicer stuff in fall? (and then should I just buy one bag of already mixed tttf/kbg?)
• I live in a development where they mow the grass for me each week and I can't really tell them not to. How badly is that going to screw me?
• My plan is to buy some dirt and seed. Place the dirt down on the bare spots, put the seed on top, then another thin layer of dirt. Use a sprinkler where I can and hand water the far back spots. That all sound right?

Any help appreciated. I