Yeah, see a filter is super beneificial for your fish! It keeps them healthy because it sustains good bacteria which eat up ammonia (fish poop and other waste) and turns it into nitrite, then that's turned into nitrate which is then taken out by water changes. But in all it keeps your fish safe and allows you to not have to change water every other day >.< it's more work to NOT have a filter! Unless you had a natural planted tank but even still, they benefit from having filters!
But yes, do some chores around the house and just agree with your mom on almost everything even if it doesn't seem quite right. Just warm up to her, give her a list but don't bug her, the bugging is what sets them off
do that and you should be good to go
And you can divide the 10 gallon in three sections (that's how mine is) so you can keep all three safe and sound!
So do you know about the Nitrogen cycle? I explained it in a nutshell up top in this message but there's a little more to it lol