First off, good luck and congratulations.
On the eggs, I assume you know that 21 days is an estimate. Due to various things, they can be a couple of days early or late. Hopefully you will have some action when you get home but even if you don't do not get too worried.
I assume you are counting the days right. You'd be surprised how many people get that wrong. The day of the week you started them is the day of the week they should hatch. If you set then on a Tuesday, they should hatch on a Tuesday. But remember that is only an estimate.
On the broody, I wait until they spend at least two consecutive nights on the nest before I consider her worthy of eggs. I've had some that faked it but were not really serious. But I've never had one that spent two consecutive nights on the nest that failed me.
I don't know where you are locarted but I give my hens eggs anytime of year they go broody. Spring and summer are better times of the year for a broody to hatch and raise chicks, but she should be able to manage any time of year with very little help from you, either hatching or raising the chicks. Her heater never has a power failure. I'd certainly give her a chance.
How many is a good question. Hens and eggs come in different sizes. A bantam hen may have trouble covering 4 regular sized eggs, while a full sized hen can cover a lot of bantam eggs.
I normally give my full sized hens 12 eggs the size that they normally lay. They can easily cover that many eggs and they can handle that many chicks too. It's not just the eggs. They need to keep the chicks warm too. There is nothing wrong with giving her fewer eggs but in winter I would not go over a dozen.
Just remember to save up all the eggs you want her to hatch, mark them, then put them under her at the same time. Then check under her once a day to make sure no strange eggs have shown up.
Good luck on the incubator hatch and with the broody.
