If your tub spigot has the little rod that you pull up to turn on the shower head, that's an easy fix - you can buy a new spigot at your local hardware store. I prefer to use a little teflon tape, as opposed to plumber's putty, because there's no danger of it attacking the plastic parts. Just wrap it around the threads a few times, and you're all set. Wrap the spigot with a rag, so the wrench jaw doesn't scratch it. This is something most folks can do with very little effort. A new spigot will run between $10 and $25.
I can't provide any answer for the water softener itself, but I do know you can install a whole house water filter, with a charcoal filter to help eliminate rust particles, and any bad taste/smell. You can get a whole house filter for around $30 to $50....there are models that cost more, but you don't need to spend that to get a good quality filter.
As for the electrical problems - The ceiling fixture is on a separate circuit from the others. Have you checked the bulbs on the sink light fixture(s)? Assuming the bulbs are good, then you may have a loose connection, or even a tripped circuit breaker. Other common electrical problems in a mobile home include crossover wiring that has become disconnected, or on the rare occasion, wiring that has been chewed through by a rodent (mice, squirrels, etc). It's very likely that at least some of these outlets are on the same circuit, in which case, its probably going to be the first outlet on the circuit with a loose wire, and because it's not working, then those down the line will not work either. A multimeter can be used to trace the circuit to find where the problem is at. I'd start by checking the breakers, then the first outlets on the circuits. Pull the outlets from out of the wall (after turning the breakers off) to check for loose/broken wires.
You may check with your local Community Action Agency, or your local Department of Family Services/Soc. Sec. offices. Typically, these folks will know if there is anyone offering services in your area. Over 90% of all Community Service Block Grants, are administered by Community Action Agencies. There should be an office in your county. You can also find help with winter weatherization, help for your winter heating bill, they can refer you to other service providers, etc. It can take some time to get registered, but its usually worth it.
Good Luck