Though germs are a concern, body odor is a very practical reason to always train with clean clothes, and for some people it requires enforcement through peer pressure.
That's just good practice in general, the salt & sweat come out easier when they haven't fully dried yet, and the bacteria won't have as much time to grow their numbers and make the garment permanently smelly.
2) also depends on the fabric, but usually you have to be pretty sweaty, at least to the wet-armpits level. Just think of the level where you will be depositing salt on the clothes.
3) Not often.
If you don't want to do laundry right away, you can soak the salty clothes, squeeze them dry, and hang them.
It would probably be wise for you to increase your washing frequency, nevertheless.
Not all blood pathogens can be killed by traditional disinfectants, even bleach. You need to soak the affected article for 10 minutes with a caustic sterilizer such as Sporicidin or CaviCide (which unlike disinfectants will kill bacterial spores)
No training center I have ever seen cleans their mats this way. Roll at your own risk...
The issue with Staph is that skin-to-skin contact, or the indirect version mediated by a mat, is more than enough for it go to "Yeah, I live here now..."
The open sore is honestly a risk for both - S. aureus loves it some open wounds.