It looks like your lawn needs fertilizer badly. It should look much nicer about mid June after you have fertilized. Highly agree that weed n feed is not a good idea. That's one of those 'too good to be true' products. It really has more unsatisfied customers than satisfied. The best way to control weeds is to control watering and to raise your mowing height. Dandelions, for example will die without doing anything once your grass grows up tall. The tall grass pushes the dandelions leaves up and it apparently doesn't like that. To control watering all you have to do is to water longer and less frequently. Weed seeds need continual moisture to sprout. If you water longer and let the surface of the soil dry out between waterings, the weeds cannot germinate. That won't control them all but once you get rid of the ones you have, then the watering and mowing height should keep them out. As an aside, if you want to show more detail in your lawn pictures, take the pictures on a cloudy day or when the sun is just going down (full shade). Here's an example of a pic in the shade...

Re: the rehab and what I did right or wrong.... 1) I would not install sod next time in the first place. I would get the exact seed that I want and call in a hydro-seeder. 2) Really make sure to scalp the lawn. I mean cut it really short! Nothing taller than 1/2". Anything taller than that and it mats down under the top dressing which means the seed won't take because the grass underneath creates an air pocket that kills the new grass, but KBG doesn't grow through like some of the warm season grasses. 3) With KBG I'd do it in May if I had a choice. There are going to be weed problems whenever you expose fertile soil, but it is manageable. Forget about the weed issues because there is an easy solution there with KBG. But KBG takes a LONG time to come in from seed. The one thing that doesn't help is if it starts to get cold before it really starts growing well. IOW, if you wait till Sept to avoid some of the weeds and it starts to drop below 50*F in Oct, it'll be till late next spring before things start to look good again. The important issues with KBG are moisture and soil temp. May/June is best IMO. 4) Do it all in one shot! If there are low spots that will require more than the suggested 1/4" top dressing, just go ahead and fill them in. Don't wait! Have your preferred seed on hand and get that thing level the first time. This work is too strenuous to be doing it more than once. Little touch-ups later are to be expected, but get it 90% done in the first shot. 5) Anything more than 4,000sf you should try to locate a mechanical spreader for the top dressing and drag the lute or chain link around with a lawn tractor. 6) This is a point of debate, but... Currently the Orbit sprinkler heads are much better than the Rainbirds. Make sure that your sprinklers are exactly how you want them before doing anything with the lawn. And yes, sand isn't great for them, but it is pretty easy to fix. There are probably more that I'll include later.

Tenacity is safe to use at seeding time, so no problems there. It can't be used after seeding until the grass is mature enough, but hopefully once will knock down some of what you have. It'll even give you 30 days of pre-emergent protection against a good spectrum of leafy weeds and a few grassy ones (most notably P. annua and P. trivialis, although pre-emergent control there is, at best, tolerable). It's getting too late to fallow and re-treat, so at this point I'd bite the bullet, seed, and realize I'm going to get every weed that the Tenacity doesn't catch. Many will die at frost. Others can be treated in spring if they come back. Watering twice a day is required for good sprout. Three times per day is helpful. Five times a day might be overkill, but it's also a way to be sure that no seed is ever dry. :-) I'd be happy if you told us you watered at 10 AM and 6 PM, though, so if you prefer that, go for it.

Weed seeds will be blown into your yard, and seeds can also be spread by wildlife. Different weeds grow at different times, Spring is usually a prolific time. A local certified nurseryman will be able to identify and recommend a herbicide. Fertilizer will not control weeds. There are companies that combine fertilizer with pesticides. I use a pre emergent fertilizer combo in the fall and a weed and fertilize combo in the spring.