Rather than turn the water on and risk getting flood damage, I normally hook up a small compressor to anywhere on the hot and the cold water lines. Let the pressure build up a little and listen for air leaks. This will tell you if you have any problems.
Since the pipes run inside of the home it sounds to me like there have been problems in the past with the plumbing. They might be perfect now...you have to check.
As far as replacing the pipes...the only way to replace the original ones that ran under the home would be to tear out the insulation under there. If you do that, you will never get good insulation back up. There is just nothing to attach it to.
If the pipes have been run along the baseboard we have boxed them in to hide them and in the front of the box have used pegboard (the stuff with holes). This allows heat to get into the box and keep the pipes from freezing because they are against an outside wall. Paint or stain the box to match the wall and they don't look too terrible.

I have repaired lots of plumbing in mobile homes and completely replaced the plumbing in two of them. Under the unit, the covering may be a heavy "tarpaper" or a woven scrim. Above that will be insulation, then the supply piping. I cut the covering and insulation as necessary to access the piping. Duct tape will not restore the covering. I re-insulate after the piping work, then use 1/4" plywood screwed to the framing to cover the opened areas. Considerable extra work and expense, but I think it is the best choice available. Carefully measure and cut the plywood for a good fit to keep out rodents.

Bus Driver...when you say screw the plywood to the framing...I can't figure out what framing. Do you mean drill into the metal framing? The only wood that I normally find is the floor joists and the pipes generally run under them.
I have tried every kind of tape to repair the cut factory plastic and have really never found anything that will do a good permanent job. Come back a year later and the repair will have come apart. Mobile home supply houses sold one called "Bottom Board Repair Tape". It just looked like black duct tape and didn't work much better than duck tape. I finally got patches from the manufacturer that were about 6" x 4' and were self adhesive. These would hold up. This is the same material that the factory uses when they have to make a repair on a new unit.

Supply piping below the bottom of the wooden floor joists in a mobile home? I have not encountered that, all the ones I have encountered have the piping above that level. In such a case, I might add blocks to the bottom of the joists to create space for the piping and insulation plus a surface for the plywood. No one said it would be easy.

I guess most of my experience has been with older ones and the piping all ran below the floor joists.