I am in the process of my second reno (different area) myself. I think it is fairly typical that some areas do not come in as well as others. Just scratch up the soil a little bit in those areas, scatter some seed and put no more than a quarter inch of compost on top of it. The existing seedlings should not have a problem with that little bit of top dressing. Just try not to smother any. I have no hands on experience with algae but I think if you just scratch up the soil a bit you will be fine.
Regarding the thickness of your tall fescue, try to resist wanting it too thick this early. What was your seed rate. About 8 lbs per 1000 sqft is the typical average rate. Some reputable sources even recommend as little as 5 lbs per for healthier mature grass stands. Once you start mowing the new grass it will begin to tiller some which will cause it to thicken up. If you get it too thick it will become susceptible to fungus problems next summer (e.g. brown patch). I think we all go through thinking it is not thick enough. When it starts to mature next year it will be thick enough and healthier.

Thank you for the words of confidence my friend! I needed that :)

Yes, there is a factor to waiting. Fescue is not very quick to tiller, but it does seem to thicken in the first year after seeding.

As for seeding rate, the recommended tall fescue rate is 8-10lbs per 1,000 sq. ft. for new seeding and 3-5 lbs per 1,000 sq. ft. for overseeding. I push the overseeding number to 6lbs. per K and if there's a really thin area even as high as 7 per K is OK.

The areas that haven't sprouted would concern me, and I would investigate the cause, and possibly reseed those areas. The thing to remember is that if it didn't the first time, it likely won't the second. Also, all a seed needs to germinate is warmth, oxygen, and moisture, so if all of those factors are there, it "should" sprout (emphasis on "should").