Can I get PTDY on the back end of a PCS?

Yes, the gaining command can authorize you PDTY, but there's no requirement for them to authorize it. It will be up to their discretion.

Yes, you can.

During my last PCS, my gaining unit required me to complete my initial inprocessing, certs and training with them before authorizing PTDY. If you have a little time, I'd make sure you have all your 350-1 certs printed, licenses, etc on hand. Then as you're inprocessing you can complete the process faster and potentially get PDTY approved quicker. Again, like others said, its up to the unit to approve, but its better to show up prepared and to give them a reason to approve it instead of being behind the 8 ball.

Yes, on a PCS Move you can be authorized up to 10 days Permissive TDY when reporting to a new unit. You won't receive travel payments or expenses during this time- but can use PTDY instead of leave at the new installation for house-hunting/moving or nearly anything PCS-related if the new Command approves & signs a new DA31.

I would call the gaining unit and explain your situation before you depart, once arrive and sign-in, you are at their disposal as previously mentioned!