How often do you water and for how long?
How high/low do you mow?

The weeds can be handled by something like Spectracide with Atrazine. But I would wait until the grass returns.

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I have gotten excellent results for fungal disease using ordinary corn meal. It is an organic approach that the experts at Texas A&M swear up and down does not work, but it has worked for me every year since 2002. It is a complete cure in 3 weeks. I am suggesting you try that because corn meal doubles as an organic fertilizer. Your spots don't look that much like a disease, but it could be. If you did not have the really healthy stand of grass right by the sidewalk, it would look more like disease. I would hit it with corn meal at 20 pounds per 1,000 square feet now and again in 3 weeks. If you don't see new grass coming up in 4 weeks, then get a commercial fungicide that works in your heat/precipitation zone - read the label closely to see how to apply and when. Corn meal has no restrictions on how or when. Then once you get the grass growing, zap the dollar weed (I assume dichondra or some other easy weed) and let it grow. And if you have not already raised your mower to the highest setting, do that yesterday.

Should I take out the dead grass first and leave the bare soil?

If it continues to grow after a fungicideis applied is a good sign it is either Take All Rot or SAD for which there is no treatment.

If it is Take All, or SAD, is it just a matter of digging up the dead grass and re-sodding the areas that need it, or does the soil need to be treated?

Here's an update. I applied the fungicide two weeks ago. I applied the Spectracide with Atrazine two days later. The dead areas seem to be filling in and getting smaller. There are runners headed into the dead spots. Also the dollarweed is turning brown and I think has reduced by at least 40%. The round leafs are turning brown but the little stocks are still green. I guess it's still working though. So far so good. Here are a couple of pictures to compare before and after:

This is 2 weeks ago
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This is this morning....2 weeks after treating
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Here is the Dollar Weed starting to die

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