Your picture illustrates perfectly the issue with using chemical pre and post M's on new installs. Chems only address your weed issue for *that* given season whether you're using a pre or post M. You can bet your bottom dollar you'll have weed germination the next year because many varities of weed seeds stay viable in the soil for a loooonnng time.

As for your specific situation......

Your grass isn't thick enough to outcompete these weeds in the picture. I'd seed some more and some more and some more. Pull a few weeds if you like but to be honest that picture looks like a fairly typical 1st year re-seed given the amount of shade Im assuming those trees are giving you.
In fact it looks a little better than some.

For the area you plan to re-do entirely, yes, it would be good to wait until fall to seed, now is probably the worst time of the year to try. If you opt for a synthetic pesticide, you would be using round-up. It works by being taken into the plant through the leaves and preventing the production of certain amino acids needed by plants - but not mammals. It is elegant in that way.

If you believe the manufacturer, it breaks down in the soil very quickly. I know for myself that you can seed an area sprayed with round-up the next day and the seeds will germinate just fine. I was startled to learn that the vast majority of soybeans produced in the US are actually sprayed with the stuff multiple times before we eat it (the crops have been genetically engineered to be resistant to the effects of round-up). Who knows how many tons of the stuff we have eaten.

So I think an application or two of round-up isn't going to poison your drinking water. That said though, I can certainly understand being skeptical of the studies that are out there, I mean, there are plenty of products that we used to consider safe but are now banned and the subject of lawsuits.

Vinegar works because it is an acid so caustic that it burns plant tissue on contact. Plants with strong root systems can survive this and come back, but not forever if you keep spraying the new growth. But I believe it only works in good sun - I have never tried the stuff myself, I have only read that. Others here would know for sure.

The best advice that I have seen to rejuvenate a bad lawn is to treat it like the best lawn on the block. If it is a northern lawn, mow @ 4" weekly, water as needed to get 1"/week all at once, and fertilize heavily in the fall with organic fertilizers.

Hand pick weeds (get a weed hound) and make and add compost when available.

I live on a well over my town's watershed. We have a nitrogen loading restriction, so going organic was easy and it really works!

Do what you are comfortable with.

THanks for the advice guys! I have a fair idea of how to proceed now.
PS: Got myself a weedhound knockoff on the weekend. It works GREAT!