I currently have a light switch in my bathroom that controls the light and also the only outlet. Can I remove the bypass the switch so the outlet stays hot or do I need to run a new line for the outlet?
I doubt anyone would intentionally wire a bathroom receptacle this way. They either made a mistake making the connections / splices, which can easily be corrected, or they ran the wiring wrong, which is more much difficult to correct.
You can correct the problem without correcting the wiring if you convert to a smart switch. If the switch box doesn't have a neutral present, this will limit your choice of smart switch, but it is still doable.
You could also remove the switch so the light and receptacle always have power, and use a light with a pull cord switch - it's perfectly sound electrically, but it's not really done in the US any more.
Yes, you can do this. If you already have an outlet, then it means you have both a hot an neutral in the box.
If I were you, I'd make sure the new outlet is a GFCI. It's not only required by code, it's also a safety issue. You can then attach the light switch to the LOAD side and it will be protected by the GFCI as well.