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Many experts feel that a certain number of weeds makes for a healthier lawn overall; also, limiting the amount of herbicide is often a good idea. This article will give you ideas on over seeding: https://www.gardeningknowhow.com/lawn-care/lgen/overseeding-information.htm

I would wait until later in the spring to fertilize, to let the seeded areas get a good hold without possibly burning the new roots of the newly growing grass. In late spring, use a fertilizer that is higher in nitrogen (the first number in the N-P-K number). Growmore has several different fertilizers, so it would depend on which you plan on using. If it has a higher nitrogen, then it will work.

Weed killers and other chemicals, when used correctly, normally stick around for about 2 weeks and then have dissipated by then so will largely not be a danger to your garden. If you want to make double sure, treat areas away from your garden by about 20'.

There's really no remedy for this other than keeping your lawn as healthy as possible. These articles should help with that: https://www.gardeningknowhow.com/lawn-care/lgen/tips-for-improving-the-lawn-and-reducing-maintenance.htm, https://www.gardeningknowhow.com/lawn-care/lgen/so-you-want-your-grass-to-grow.htm

Spring is a good time for planting warm-season grasses but you will still need to keep it watered over summer while it establishes strong roots. Generally, however, fall is better for planting most other types of grass seed. Fall weather has an ideal amount of rainfall and temperatures for grass seed germination. You have two options for fall grass seed planting. You can plant the seeds in September and the grass will germinate and establish this year, or you can plant the seeds after this, and they will stay dormant for the winter and germinate in early spring, when the conditions are ideal as well. The bonus of a late fall grass seed planting is that many of the local birds will have left by then, making the seed less vulnerable to them (as spring planting means also protecting the seeds from birds). Then the winter rain or snow will quickly cover it up until spring.