Mimosa are a weed here and are just as comfortable in a standard potting mix as to peat/sand or clay, so I doubt they care, they also infest many crops here, so fertiliser seems to have very little negative effect on them, if any, I cant say for sure (as I don't live in your climate) but I would imagine a shallow water tray, tap water should be fine and potting mix you have spare, they seem to colonise gravely clay faster here, so maybe a wet open mix.
M. pudica is a heavy feeder. Any run-of-the-mill potting soil you can buy at the store will work. Also grab some miracle-gro while you're there because they will need that eventually. Warm. Bright sun. The healthier you grow the plant, the faster it will move and "recharge" the leaves.