Lawn#1- If spraying the crabgrass will potentially get some overspray on the freshly seeded area I wouldn't do it. The new tender shoots could easily be damaged by the herbicide. If you can spray the crabgrass without getting it on the new grass then by all means go for it.

Lawn#2- On the fertilizer either way will be fine and IMO really won't make any difference. I would wait on putting down any lime as it could affect the fertilizer. You really don't want to put down both at the same time. It's best to wait a month or two if possible to put down the lime.

Crabgrass should go away on its own as the weather cools. It will start turning purple.

You could maybe try using one of those new RoundUp deodorant sticks on crabgrass and weeds. That could help avoid overspray into your freshly planted grass.