You did not mention watering but you're going to have to water weekly in the summer no matter what kind of grass you end up with. While bermuda will survive going for months without water, it will become weedy if you try that. Many people are dead set against watering their lawn. If you are one of them, then you may as well just mow what you have because it will all become weeds without controlled watering.

Bermuda grass does not die in the winter. When it is cold enough and it gets no water, then it will go dormant. If it does not get cold and you continue to water it weekly, it will remain green all winter. Many people disagree with me on that point but I see green lawns all winter in my neighborhood in San Antonio. Other lawns are dormant, but the ones getting weekly water remain alive and growing. Plus, rye is a bad idea for the overall, long term health of your bermuda. Sports fields can get away with overseeding with rye but they are allowed to use chemicals in the spring that the typical home owner cannot get.

I looked at the rest of the pictures on the imgur site and nothing else. Is this the front of the house?
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If that is the house, is it my imagination or is there a mound of dirt at the left corner of the house? Maybe not. If there is that has to get leveled manually.

See the rocks up against the house and see the weeds growing through them? Those rocks must go or you will have a weedy looking front forever.

If you have a concrete blob under the soil surface in the back, dig that out manually. It happens all the time that construction guys think they are getting away with dumping their excess materials on the owner's yard. They're long gone, so dig it out yourself. Nothing will grow there.

You have too much shade to grow a nice, uniform looking bermuda lawn. This is true in front and back. You can try it but you'll be fighting a losing battle. In the shade it will thin out and you'll be reseeding bermuda every summer trying to fix it.

The only alternative for that much shade is St Augustine. St Aug is a coarse bladed grass which is mowed at the highest setting on your mower. It can be mowed lower, but it becomes weedy when it is lower. It also uses more water when mowed low.

How level is the ground now? When you dig out the concrete blob you can scrape soil from your high spots into the hole where the concrete was. Be sure to mound up the soil there so it will settle more even. How much you mound it will depend on how wide across the hole becomes. I would mound up about 1/2 inch for every foot across the hole becomes. Use nothing but soil to fill that hole. If you use compost or other organic stuff, then all bets are off. Compost completely disappears and has zero long term volume. Use sand or your local topsoil.

If you are patient you can convert your entire lawn to St Augustine for about a dollar. That's how much one flat of St Aug sod costs. Place that down and start watering it. It will spread 5 feet in all directions once it knits into the soil and the fall temps cool down. St Aug spreads twice a year in the spring and fall. It goes about 5 feet each time if you are watching it and keeping it moist. When you water it weekly and mow it at your mower's highest setting, it will take over where the bermuda once was. It is one of the few grasses that can chase out bermuda. The reason it works is the same reason the rye hurts bermuda in the spring. The tall grass provides enough shade to weaken the bermuda. Eventually the bermuda goes away. I've done this many times in my yard.

If you are less patient, you can put down more flats of St Augustine or cover the entire yard. I would get the Floratam variety if it is available. It performs very well in the full sun and pretty well in the shade. There are other varieties. Ask about each one at your local nursery. Ask to see some installations where the yards are mature.

I would not get into a hurry to put in a sprinkler. Instead get one of the good, turbo oscillator type sprinklers. They are excellent for a rectangular shaped yard. The turbo types usually cost about $20 but might be less this time of year. Turbo oscillators last a couple years and give a very even and precise pattern. Also get a good Craftsman hose. Craftsman hoses from Sears are unconditionally guaranteed for life. I have seven of them (two houses). Any time the dog chews one up, they exchange it no questions asked. I've exchanged one of my hoses 3 times.

Also don't worry about prepping your soil for new grass. First get the new grass growing in the soil you have. Once you have roots, then you can start fixing things. I have been fertilizing with organic fertilizer since 2002. Once I started that, almost all my lawn issues went away. I would suggest starting with alfalfa pellets (rabbit chow) or soybean meal. Both go down at 20 pounds per 1,000 square feet. You can try using chemical fertilizer if you like, but I found it was too easy to make a mistake with that. With organics if I don't use enough, I can always add more. If I use too much, it just lasts longer.