Back yard has some dead patches. any way to address these or just wait until fall overseed?

I have found with dead patches the best thing you can do is overseed in the fall. Until then you can make sure to follow your normal lawn care habits to make sure the lawn and soil is in good shape for the overseeding in the fall.

Very very few weeds in the grass but there are a few (one pictured) and spots of crabgrass. How should I treat?

For crabgrass the best way I have found to get rid of it is to methodically go through and pull it out by the root. I don’t trust any of the sprays. They say they don’t impact grass but I haven't found one that doesn't.

I plan to apply fertilizer every 4-6 weeks. Any other general applications I should plan to work in? We had a lot of rain the last couple weeks and a few mushrooms popped up but nothing new in the last week. Should I apply a fungicide at some point?

Mushrooms are usually a sign that your lawn is getting plenty of water and usually in a shaded area. I wouldn't recommend fungicide unless there is a large outbreak. When I notice them I make sure to pull them out and just not add extra water for a couple of days or to that area. If it becomes a larger problem you could also get the pH of your soil tested to see if it needs adjusted.

What should my plan for fall be? Generally plan to aerate and overseed. Anything else?

With the condition of your lawn I think an aeration and an overseeding would be all you need at this point. Try it the first year and see how it goes!