Can (and should) I replace the hose connecting a cold water expansion vessel to the cold water supply myself?
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.