Mohawk and Sahara (didn't know any better when I planted it) are two of the varieties of bermuda that I know are there, because I planted them.

Last time I mowed the lawn I lowered the cutting height to about 1.5 inches from about 2.5 inches. Now there are many areas where the brown/tan stems of the bermuda are showing.

I'm confused by some of the previous threads on this subject. Will the tan stems produce green leaves at this height or have I cut it too low? I also have quite a weed problem, including a bad infestation of Virginia buttonweed (brought in with a load of fill dirt), but it's too hot to apply herbicide now, which is why I'm trying to cut the weeds as low as possible so that the bermuda can eventually choke them many of the weeds out.

Should I fertilize now to encourage the bermuda to grow or will that just be soaked up by the weeds as well?

What has happened is your Bermuda was trained by you to grow tall, rather than lay down. Now you have to train it to grow at a lower height. By doing this you will cause the Bermuda to grow much thicker and denser which will help choke out the weeds.

But there is a catch. You can maintain your Bermuda down to a 1/2 inch if you wish and have a reel mower to cut that low. Bu there is the catch to go that low requires mowing just about every day or at most every other day. You cannot remove more than 1/3 of the height. Maintained at 1/2 inch requires mowing again at 3/4 inch. Bermuda can easily grow 1/4 inch per day.

If you have a rotary mower, most likely the lowest you can go is about 1-1/4 to 1-1/2 inches which is just fine for an improved common type you have. Just keep in mind it will require mowing twice a week at that height.

So pick a target height, read the Bermuda Bible, and follow the instructions. Just be aware the height will determine mowing frequency.

Don't worry about the cool season grass, it will be toast if you water, feed, and mow Bermuda properly. Fescue will roll over and dies when cur short, heavily fertilized in the heat of summer with with only 1 inch of water per week.

There's about 30,000 square feet of lawn here with the mixed varieties described above, some of it under high filtered shade from tall pine trees and I'm wondering if some of the newer seeded varieties like Yukon or Princess might be more appropriate in the slightly shaded areas? Is there a way to plant (overseed) the newer varieties so that they might overtake the Sahara/Mohawk without starting over from scratch?

This lawn is in the metro Atlanta area and the soil here is mostly Georgia red clay. I understand the red color comes from iron content and since iron is a recommended nutrient, can I release some of the iron from the clay by applying lime to raise the pH, which is typically around 5.5 in this area? Will applying lime in this sweltering heat damage the bermuda?

Or should I apply a fertilizer with iron in it at the appropriate time?

You can overseed other areas. I did that in my back yard with Triangle. You do need to fix the pH, but it takes time.

I do use iron a couple of times a year. TW recommends the Bonide liquid product and it works great. Pike caries it.
But I think the pH comes first...

As for iron chlorosis and soil PH, it is high PH soils (7.0 and higher) and presence of lime that locks up iron and prevent iron uptake. You said you have a soil PH of 5.5 (acidic) with heavy red Georgia clay. While your PH is a huge life long loosing battle of a problem. it is the heavy clay and soil compaction that will cause iron chlorosis. It prevent oxygen (air) penetration into the soil. The liquid iron products help because the iron is absorbed via leaf surface, but is only a band aid and short term solution. The real solution is aerating and loosing up the soil so it can breath.