I have never had braces. As I get older and my teeth move, it may be inevitable.
Yes, it might be if you are determined to keep your teeth in line.  Orthodontic cases align and straigten teeth
My front teeth are beginning to come together, and I have a gap in the back of my mouth where I had a tooth pulled.
Yes braces can correct that, overbites, underbites, etc
I wonder if they would correct teeth grinding at night?
You might need orthodontic headgear for that.
What is orthodontic headgear? That sounds horrible for sleeping!
Yes, it is.  It attaches to your head with a neck strap