1) From your description it is hard to tell what is wrong with the camera. I recommend taking it to a camera shop and showing someone there.
2) As the saying goes "The best camera is the one you have with you at the moment."
A *good* camera depends on what you are intending on using the camera for. If you are shooting friends or taking pictures when you travel once a year, then a Canon Rebel (or Nikon equivalent) is great for your needs. If you are trying to make a living as a commercial photographer then I'd recommend a Canon 7D, 6D, or 5D. Along with the camera you will also need to invest in lenses. Lenses often make more of a difference in image quality than the actual camera as most sensors are relatively the same in most cameras, only the amount of pixels changes. But high end lenses have very nice glass in them which will allow you get very sharp and clear images, whereas the cheap kit lenses that come with cameras are lower quality and will never be as sharp.
3) Again, it depends what you want to take photographs of with your phone. The Google Pixel 2 is supposed to have a really good camera on it. Samsung Galaxy also is known to have a really nice camera (many people I know who have tried both prefer the Galaxy S7 or S8 cameras to the iPhone 7's camera). I have not tried the iPhone 8 or X, but those are rumored to have great cameras as well.
4) Personally, I shoot things, then back them up on my home computer as soon as I can. I leave them on my home computer for no more than 6 months before moving them to another hard drive. You can buy 500GB laptop hard drives for relatively cheap. These are great as they do not take up a lot of space on a shelf or in a drawer, but hold a lot of data. Every 6 months or so I dump the images that I am not working on from my home computer to a hard drive, and then put it in a fire-proof box. (Even better would be to store them off-site somewhere safe). if you want redundancy you can have two backup hard drives, or burn DVD backups, and then copy to a hard drive as well. This way if a DVD fails, or a hard drive fails you have a second copy.
hope this was helpful :)