1) The following tickets require a handstamp for re-entry/park hopping: 1Day1Park, Park Hopper, Park Hopper +, Ultimate Park Hopper. The stamp is used to prove that you were in a park previously and will be checked if your ticket has been used too recently or if the turnstile computers go down.
2) The following tickets do not require a handstamp: FL Resident Play 4 Days, FL Resident Seasonal Pass, Annual Pass, Premium Annual Pass. These tickets utilize biometric finger scan to verify the ticketholder's ID. In the event that the biometric scanners are off-line, or if the turnstile computers go down, valid photo ID is required to verify ticketholder information regardless of whether you have a stamp on your hand. If you're using Bob Smith's PAP and you're not Bob Smith *or don't have ID to prove it* it doesn't matter how many stamps you have.
3) There are other ways for CMs to verify the ticketholder, but they will only be used if the system deems it necessary. 9 times out of 10, your stamp will not be checked. That does not mean that the stamp is just for show or that it will never be checked, it only means that 9 times out of 10 verifying the user is not as high a priority as getting people through the turnstiles in a hassle-free manner.
The ticket is the most important part of the equation. Make sure you always have it. Get a stamp Just In Case (it doesn't hurt, stain, cost anything, or take any major amount of extra time or effort, so why not.)
The ticket will work multiple times in the same day.
There are many reasons a ticket might not work in the turnstile. If they send you to get a new ticket, it has nothing to do with the stamp or the biometrics. If your biometrics do not match or if you don't have a stamp, that has nothing to do with the mag-stripe on the ticket.