It would be helpful for you to know what is the thickness of the bermuda sod that you want to install. If you know where you are going to purchase the sod from, call them and ask them what is the depth that their sod is cut. It is probably going to be somewhere in the range of one-half inch to one inch. Dig out your area to the same depth as the sod is cut. Use a ruler or tape measure to measure the depth of the area that you dig out. Then the sod you install should fit nicely. If anything, it would be better to have the sod a little lower than the existing area rather than a little higher than the existing area. This will prevent scalping of the new sod. Of course, perfectly level is best. :)

Not sure I see the issue. You mow grass, including sod, when it is at a height where you are going to remove 1/3rd of the blade. So let's say you are going to mow your grass down to 3 inches. When the grass is 4 to 4.5 inches, you mow it. If the old sod is almost at that height, then I doubt the new sod is. So if you go over the whole yard, the newer sod will be under or just at the mowing height, and the blades won't be cut, while the old sod blades are cut to the proper height. Does this make sense?

Next time you redo your lawn, (1) please do not till, (2) roll the seed instead of raking it, and (3) please consider mixing Kentucky bluegrass in with the fescue. Because you tilled, you will have to relevel the yard after the fluffy tilled soil settles. The settling process can take 3 years, so don't get excited and try to fix that now. Just remember to do it then. Because you raked the seed, anything could have happened. I understand your motivation, but rolling works to do the same thing without moving seed around. Because you seeded with fescue and raked, you will have bare spots. Fescue grows one plant from one seed. Each fescue plant grows outward at a glacially slow pace. If you had planted a mix of fescue and Kentucky bluegrass, the bluegrass would spread like a carpet in and around the individual fescue plants to fill in all the bare areas. Because of these different characteristics, fescue is called a bunch grass and Kentucky bluegrass is called a sod-forming grass. If you want dense fescue you have to plant a lot of individual seeds. If you want dense Kentucky bluegrass, you need plant only one seed and wait. Of course it is customary to get better coverage of KBG by planting it like any other seed (lots of it). So, getting back to your question, YES, reseed as soon as you realize you are going to have bare spots. Assuming the bare spots are relatively small, you can achieve the same effect as rolling by simply walking on your seed as you seed it. Your full weight on flat soled shoes is perfect to compress (not compact) the soil with seed on it. Mother Nature seeds and "rolls" by running livestock over it, so your feet aren't going to hurt anything.

Well for one your watering is not right. But I like the shade problem, too. Most grasses need full sun to thrive. On a mature lawn you should be watering more like once every three weeks instead of twice a day. You could easily have a disease that is taking out the grass. The watering mantra is deep and infrequent. Deep means 1 inch all at one time. Measure an inch by placing cat food or tuna cans around the yard. Put them in the good areas and the bad areas. Then time how long it takes to fill the cans. That will be your watering time from now on. Infrequent means you allow the soil to dry completely before watering again. With temps in the 90s you should water once a week. With temps in the 80s water once every 2 weeks. With temps in the 70s, water once every 3 weeks. That should cover it for you unless you live at altitude. By watering this way you discourage the fungal diseases from getting started. You also discourage weed seeds from sprouting. Your current watering is what is recommended for sprouting new seed. This watering schedule works everywhere with very few exceptions (Las Vegas sandy soil mixed with blazing heat means they might have to water every 5 days in the hottest part of summer). In HI you might need to stretch out the frequency due to humidity keeping the moisture from evaporating out of the soil. Rain counts, too, so if it rains, restart the calendar. What was in the seed you put down? There is a Guaranteed Analysis on the bag or box. Would you be interested in investing in a new mower with some flexibility in mowing height? Most grass does best when mowed at 4 inches. Bermuda and centipede do best mowed at around 1 inch. Images always help. If you take a picture, please do it on a cloudy day. If you want to take a picture of the dead grass near the fringe of the bad/good spots, get down to 3 inches away so we can see what the individual blades look like. Also you might want to look into organic fertilizer. The simple way to do that is to visit your feed store and get whatever is cheap. Usually that is corn meal (if you live in the states) or alfalfa pellets (rabbit chow). Scatter that around at a rate of 15 pounds per 1,000 square feet. Here is a link to feed stores near Honolulu.