Keeping fish alive on stringers isn't specifically mentioned though live wells or tanks are not allowed - you can't keep fish alive in a well or in a bucket or other container.
Stringers are actually mentioned in the regs in the context of keeping live fish on them (ie dont do it) but since one is not allowed to have any live fish in their possession at all, it's kind of moot.
3. If you want to keep the stocked trout, do you have to kill it immediately and no stringers are allowed?
Basically any retained fish of any type has to be dispatched immediately, but then you can use all the stringers you want.