I'm sorry you are learning the hard way that breeding budgies is not as easy as "just put mom and dad together".
I've been breeding budgies for a very long time (and have just decided to retire) so I'll be happy to walk you through all your questions, and even more questions you'll have along the way.
Like Faery has already mentioned, you can remove the egg shells when the hen pops out of the nest box.
You will need to check the nest box twice a day as the eggs are hatching. Check in the morning and afternoon to make sure all is well.
Once all the eggs have hatched, you'll need to start cleaning the nest box. When the chicks are small, you can do this every few days, then as they grow you'll need to clean it every day, and then likely twice a day once they get really big. (2.5-3 weeks old). You want to make sure that you are keeping the nest clean and dry so that no bacteria can grow and harm the chicks.
Use a safe bedding like pine shavings or aspen shavings. You can find these at any pet store. I've also seen people use dry oats as bedding (although I personally have never tried this) if you can't find wood shavings.
You can start handling the chicks for short periods of time once they are around 2-2.5 weeks old. Make sure you wash your hands, and that your hands are not cold. You can hold each one for about 15 minutes at a time, and as they grow and feather up, you can extend this time little by little. Your chicks will then be used to you and your hands and will be hand tame.
