You can replace the wheel. It looks like you can get a bolt on derailleur hanger, supposedly. I've never used those. You need to measure the dropout spacing on your bike frame. 135mm is the normal mountain bike size. 130mm is the road bike size.
The other issue is what you will do for brakes. I am not overly familiar with your frame. I don't know if you have mounting points for it. The best I can tell is that you don't. I also don't know if you have cable stops. You have coaster brakes, as best as I can tell. Mountain bikes don't use coaster, so you will not be able to get a coaster wheel unless you have one specially built for you. You can try to put on hand brakes, but I can't tell if your frame has mounting points for them.
Since you are not knowledgeable in bike mechanics, you'll pay someone to do the work for you, which will run you a pretty penny for parts and labor. You'll be getting close to new bike territory.
The Mongoose is a Walmart bike, I would pass on it.
You can do a CL search with a maximum price limit, and search for Specialized or Trek, which are the popular brands. You're looking for an older bike, with a rigid fork.
Specialized Hardrock That is an example. It looks a bit on the larger side, over 17" frame. It could be a 19" or 21", hard to say, and may not be available anymore.
Here is another, though I don't know much about older GT bikes.
GT Outpost Trail
And another
Trek 830 Antelop
Edit: These old bikes take a quill stem, which allows you to raise the handlebar height quite a bit. The newer mountain bikes use threadless forks, so you are much more limited. Hybrids (new) still uses quill stems.
Take someone who knows how to work on bikes to give it a once over before buying anything.
If you're not comfortable with that, then your only real option is to buy a new bike. In that case, see if you can find some new hybrids on sale. The Trek 820 won't do it for you, it's not upright enough. Something like a Trek Pure would fit, but it's out of your budget unless you find a sale. I don't shop hybrids so I have no real recommendations.