Actually, I think the label tells you not to apply it right after mowing. The idea behind that is that you want to have as much leaf as possible to increase the amount of weed killer that gets absorbed. What kind of week killer did you use?

You need to mow. Seedlings at 1-2 inches are pretty tough, even though they don't look it. I would cut the grass to about 4 inches ASAP, then in a few days, bring it down to 2.5-3 inches.

A weed killer is a plant killer and if you apply it so that there is a chance that it will get into your grass via the freshly cut grass blade, then you might be setting yourself up for some grief. Which is the long winded way of saying 'wait a day or two to ensure that the cut ends seal'.

I am by far not an expert grass guy but I can give you some insight into what I have learned after starting 3 large lawn areas under less than ideal conditions. It sounds like you have done a great job so far. IMHO I would mow your grass will be 2 feet tall if you wait for it all to germinate and be 3" tall. Sure your going to crush/damage/kill some seedlings but how many are you going to lose if they are deprived of sunshine? Plus you will be stressing your existing grass by chopping too much off. The clippings, if anything, will help retain moisture for them. I try not to step on or roll over known seeded areas, sounds like this might be hard for you. I agree with waiting 8-10 mowings to broadcast spray. I have killed seedlings by jumping the gun. I personally would wait till fall. You could spot spray and will only risk killing seedlings with the over spray or, thanks to a user here, use a small paint brush and paint each one. I over seed my lawns every spring and fall. I broadcast spray with killex the day before I am going to seed. Watch home depot and the like in late fall. They usually blow out seeds and other gardening stuff at pennies on the dollar. One thing that really helped me and my lawns was buying a soil test kit. I test my soil about 3 times a year (spring, mid summer, fall) to find out what it is lacking and buy fertilizer to balance it out. I didn't realize my soil was very alkaline and after dropping the PH my lawn is doing WAY better. Good luck SCG