I don't have experience with an STP packer but I figured I'd try to answer some of the questions if I could.
Period:
I never wore my packer during my period, as I was too afraid to get blood all over it. I washed it thoroughly about every few days with hand soap, but I get so dysphoric about the blood that I didn't want it all over my penis. If I were ONLY packing, not STPing, I would think about wearing briefs with a pad (as I too will never ever put a tampon or a cup up in that area) and then wearing a second pair of underwear in which I would put the packer. I would say a period would make it very difficult to keep an STP clean and probably wouldn't recommend trying to use it then. If I suddenly got my period, I usually was light enough I could just wait until I got home and then remove the packer.
Packer Care:
For the Mr. Limpy, there's no using it for penetration or anything like that. (Though I've found that packers can be nice for masturbation, if you imagine that it's your own penis, but that's beside the point here.) If you use a harness, if it's got an O ring or anything like that, watch to make sure your packer doesn't begin to detach from the balls. Tears in that area are common. If you swim with it, clean it when you are done. If it's sticky, corn starch does wonders I hear, though I never had a sticky packer myself.
Sleep and Shower and such:
I am almost always wearing my packer. I wear it to sleep, I bring it into the shower with me, I wear it during the day, I only take it off to clean it and if I'm feeling sensitive down below that day.
If you're curious, the packers I have used are the MASHO, the Doc Johnson Light, and the ReelMagik Basic Packer. I currently only use the RM, as the masho is too small for my liking and the Doc Johnson fell apart. Also the RM feels much better. I do have experience with a medicine spoon STP, however, not within a packer. I was fairly good at using it without clothes on, however, once clothes were added, I struggled to position it correctly. It was just my anatomy, I know many guys do have success with it and love it. I plan to buy a peecock soon, but right now I have to pick between my name change or the peecock, and my name change is more important.