Sorry it took so long for someone to reply, i hope you still seek answers :)
I have been breeding budgies for 2 years now and each time I get excited when my new lot of babies arrive!
Your first question is hard to answer as I have never known or seen any of my budgies mate...I simply look in the nest box and Poof an egg is there!
From my experience I usually look at the females vent (bottom) to see if it is enlarged. What I have found out is that it is bigger around the lower belly area on a bird before she lays an egg. also her poo is a tell tale, it will be quite big and very 'soopy' as she is getting ready to lay. If you see this then expect an egg within 1-2 days.
The eggs take 17-20 days to hatch from when she starts incubation. She may start sitting once she lays egg number 3, this is fine. All my budgies start incubation on egg number one. So i keep track of the days from there. Usually the babies hatch on day 18. Do not panic if a baby hasn't hatched after day 20. One of mine didn't hatch till day 25, and is a healthy and happy budgie today! You can also candle the eggs to see how the baby is growing in the egg. This is super easy, you simply get a torch and place it next to the egg and see your baby growing inside! On day 4 of incubation you can see a heart beat.
I hope I have helped and answered any questions you may have wanted answers to :) either way I'm happy to share my experience!