There are several issues here. I am only going to address a couple. I know some experts will give you more advice.
Firstly, budgies actually like noise, as silence in the wild is a sign of danger. One way to help your budgies settle in and feel more comfortable is to sit by the cage and talk to them and read to them. You can also have music playing. They often find this calming and may even start chirping.
Do NOT starve your budgies into submission! Feed them the proper amount of food every day. There is no reason you need to get them finger tame right away (some never become finger tame). There are some great articles on taming and bonding in the subforums that can help you learn how.
Yes, it is not good for your mom to be sticking her fingers in the cage. Budgies are naturally afraid of hands to begin with. You want want to have your mom read some articles about budgie care so that she knows why you don't like it when she does this.
As much as you can, create a safe space for your birds, and they will be much happier. I can tell you care about your birds and want the best for them. With some learning and a little work, I think you can create a good home for your birds.
The very best advice is to read all the stickies ("stuck" to the top of the subforums") and articles on here. You will find lots of excellent information.
Good luck with your budgies!
Goldenwing (and Lemon Drop)