While the lime trick may be working for you, there really isn't any agronomic basis for lime directly combatting weeds, especially with frequent applications. With too frequent of applications, you are going to raise the soil pH too high, which is not necessarily too bad for most turfgrass, but it does promote a hospitable environment for clover to grow like crazy (it likes a high pH). On a related note given the site we are on, wood ash is an excellent liming source!
Here in VA, clover, dandelions, crabgrass, goosegrass and chickweed are our main turfgrass weeds. In addition, we fight bermudagrass (wiregrass) in our cool-season (bluegrass, fescue) lawns as well. Most any of the over-the-counter weed killers containing 2, 4-D and/or Dicamba will zap most of your broadleaf weeds (clover, dandelions, chickweed). Removing the others is trickier - there are several pre- and post-emergence crabgrass/goosegrass killers that work moderately well, with the pre-emergent ones generally working the best, although a combination will probably be needed for season-long control. If you have heavy bermudagrass and/or other weed infestations, your best bet may be to go ahead and zap the entire stand with a round-up type product (glyphosate) or other product (gramoxone) that kills everything, and start all over.
Ideally, the best time for seeding and fertilizing cool-season grasses is in the fall, specifically 4-6 weeks prior to first killing frost. Temps are cooler and moisture retention higher during this time. Seeding and fertilizing can be done at the same time as stated in earlier posts. If you have no seeding to do, it is still best to fertilize in the fall - spring fertilization is for those that enjoy recreational mowing and overseeding their lawn in the fall after half of it dies from overgrowth/drought stress during the summer. If you have to seed at other times of the year, try to use a ground cover like straw to retain moisture and shield from excessive heat, and keep the new area watered, but not overly so.
Do not attempt any overseeding for at least 6 weeks (or longer depending on label instructions) after applying a pre-emergence herbicide. The "kill all" herbicides (round up, etc) and/or post-emergence herbicides can generally be seeded into within a couple of days, but again, make sure and read the label as different products have different restrictions.