From the way you describe it, it sounds like it's a regular gate valve. You might be able to repair it as you describe by removing the packing nut and replacing whatever is broken or stripped out inside.
The only exception to that would be a ball type valve, but those will just have a handle that makes only a 1/4 turn. Those generally aren't repairable, but then again they almost never fail either.
Personally, I would just remove and replace the gate valve with a good ball valve. Gate valves have a tendency to rot out or seize up from lack of use. I actually had one once that disintegrated in my hand when I tried to turn it off (spraying water everywhere and making a nice mess). Ball valves are just much more reliable.
I'd try to eliminate the flexible hose if at all possible. You can get a threaded piece of copper pipe and solder it into your plumbing system if you have copper plumbing.
In a well designed system, all of the hoses, valves, etc should exceed the threshold for the TPR valve on the hot water heater. So maybe the hose you had in there wasn't rated for enough pressure, or your TPR valve isn't functioning (very dangerous, you should test it).
For the tank itself, don't worry about water coming out of it. It will work just fine without water in it (in fact, it will likely work better since you've increased the volume of air absorbing pressure fluctuations). To pressurize it, you have to remove all water pressure first (shutoff the water supply and open a faucet) and then pressurize it with a pump up to your water pressure PSI level. As long as the tank holds pressure with the water turned off, the bladder is still good, and you can reuse the same tank.
If you're worried about dirt in the water side of the tank, fill and drain it a few times to see if you can clean it out. Chunks of rubber will likely get stuck in an aerator or shower head which would need to be cleaned out, so better to do this now.
In terms of mounting, they can be mounted any direction. If mounted vertically on a fixed pipe (doesn't matter which end is up) then you don't even need support brackets.
Best Answer
As long as you can get the existing valve fully open you can leave it in place. Buy a piggy back valve, you screw it into the existing valve and use it instead of the original.
As zelinka mentioned the existing valve is done in expansion pex and the tools needed to put a new one on are close to $600.
You could probably do a sharkbite fitting on the pipe if you cut it below the expansion ring. Something like this:
https://www.homedepot.com/p/SharkBite-1-2-in-Push-to-Connect-x-1-4-in-O-D-Compression-Chrome-Plated-Brass-Quarter-Turn-Straight-Stop-Valve-23337-0000LF/202721912