Have you tried Kegel exercises? They can really work for situations like this. Just clench the same muscle you use to stop your urine. Do several of these at a time 3 or 4 times a day. You can do them anytime, anywhere because no one can tell. They will strengthen the muscles so you have more control and tighten things up a bit, too.
It also sounds like you need to air things out "down there" more often if you are getting irritation. Are you using only all cotton pads and underwear? Are you making sure only to use gentle, unscented products (including the pads)? Maybe after you go to the bathroom at home, you can take off your undies for an hour or so, to let things breathe. Some cream on the irritation would help, too.
If your cough doesn't get better after a couple of weeks, I would defiantly see a doctor to make sure you don't have a persistent infection that is making you cough.
Hope you feel better soon.