You're conflating coils and contacts. A coil is what you use to change the status of a binary memory bit (0 or 1), a contact is what you use to perform an action based on the status of that bit. (apologies if you already know this but your comment indicated to me that you might be confused on this point)
"Setting" a coil is the same as latching it. It will stay on until it is unlatched (Reset). A positive transition contact will pass logic power flow for one PLC cycle when the the coil is energized (whether by direct or latching). That positive transition contact will not ever pass logic power again unless the coil turns off then on again.
It absolutely does (if you coded the button correctly), but with a direct contact instead of a positive transition contact, it can execute many times (maybe hundreds of times) before the reset occurs because of the time it takes to remove your finger from the button.
No, absolutely not.
Yes, correct.
I don't know what you're asking here.