Sorry about your Mom....
You should back up your phone to iTunes right away. If you do not have it backing up to iCloud, and you are not backing it up to a computer, you risk losing everything. With no backup, you have no way to recover your contacts that have been lost.
Your photos are even more important....back them up! As you said, you don't want to lose them, and the only way to fix it is to have a backup.
You need to hook up your phone to a computer that is running iTunes, and that has been authorized with your AppleID. Here is an iTunes primer if you aren't familiar with that whole thing:
Once iTunes is installed on your computer (Windows or Mac), then you would hook up your phone to the computer using the USB cable that came with the phone so you can sync it:
Once you have sync'd your phone, it is a good idea to do so regularly if you are not planning on using iCloud. If you decide to use iCloud (which is free for the first 5 GB of storage), then much of your stuff will be backed up on the Cloud on a regular basis, and the need to sync to iTunes becomes less urgent. Here is the iCloud primer in case you are interested in persuing that:
Hope this all works out for you. I'm sure this is a really hard time for you, and to have to go through a whole new learning curve at the same time is going to be difficult. But hope you are able to get through the basics so that you will at least have the peace of mind of knowing that your stuff is backed up and retrievable!
Ah yes - the Avatar pic....when I first started helping on the forum, I wondered the same thing!!! As you probably have noticed, various people have different point levels. You get points for answering questions correctly (10 points) or helpfully (5 points). It's called the "Reputation System" and you move into different levels with different privileges. So, once a person has racked up 500 points, they reach level 3, and they get to add their own custom Avatar....
I actually liked the "generic" one I used up to that point...I had the rubber duckie one, and I really thought it was cute. But I'm glad to have my kitty representing me now!