Wow!! I can tell from what you have written you are a caring young person who is trying to do the best thing possible for your cat.
First off - please do not allow anyone to take any kitten from the mom until at least eight, preferrably 10, weeks. They are learning the important role of socialization up until then as well as getting their very important nutrition from their mother.
Secondly - It is going to take some time to get back to "normal" for your cat. Having someone trying to take one of her babies is very traumatic. I can't understand why your mother would allow that to happen. Please DO NOT allow that neighbour to have a kitten. It would end up dead very quickly.
Now - things that may be beyond your control but they are vitally important -
the "chip" that is put in is NOT a "travelling chip thing." It was inserted so you could find your cat if she ever escaped from your home. It sure isn't intended for your cat to be able to roam the countryside freely. Normally it is done at the same time as a cat is spayed/neutered.
Please get your cat spayed as soon as possible after the kittens have stopped suckling. That is normally a week or so afterwards to give mom's milk time to dry up. Please do not let your cat outside!! She will just end up pregnant again. The last thing this earth needs is more cats! There are millions being euthanized every year because there are no homes for them.
Please do not "give" kittens away. You should be trying to charge a minimum of $20 for them. You are essentially ensuring your kitten is going to a home that may look after it well by doing that. If you give them away free they could end up going to a lab (many people go around looking for free kitten signs and then take the kittens they get to a lab where they get paid for bringing them) or being used for snake food. It happens on a regular basis.
Please, I beg you, ask your mother to find a new vet. That one should not be saying your cat is not going to be aggressive for the rest of her life. For Pete's sake, she was defending HER children!
With care, time and patience she will return to the loving cat she was.
Let her raise her babies, get her fixed, then spend the time to reassure her things are back to normal.
Almost last - PLEASE do not let your cat outside!! There are far too many dangers for cats outside. Cars, poison, people who want to harm them, coyotes if you live in the country, other cats who are sick and can pass on diseases, not to mention fleas and other parasites.
Lastly - ask your mother than no boxes be moved around the cat. Clearly she looks at them as a threat.