For house hunting + PCS leave, I need to report to Housing Office first. If I take PTDY first, do I need to show up after the 10 days?

This can be confusing, but your reporting to the Housing Office and the signing of your lease will document the start and end dates of your PTDY. Your leave form will document the start and end of your leave period. If you sign in to your new unit after having completed both your PTDY and your leave, everything gets a little muddied (the PTDY will be contained within your overall leave period, and will require an adjustment to your chargeable leave). This is more complicated than signing in, and then going onto PTDY at your new unit, so be careful and make sure you handle this correctly.

You'll need to demonstrate that you were actually in the area looking for a place to live in order to get credit for your PTDY (and not charged leave for those days). That's done by first reporting in to the Housing Office (to establish the start date of the PTDY), per the regulation. Then, you provide the Housing Office with a copy of a lease showing when you found a place to live (to establish the end date of the PTDY). That time period will be your PTDY, and all other days on your leave form will be your chargeable leave. If it only takes you 3 days to find a place and sign a lease, you'll only get 3 days of PTDY (and the rest will be charged as leave). If it takes you 10, you'll get 10. If it takes you 20, you'll get 10. If your search occurs while you're still within your leave period, your leave doesn't need to be "restarted", because you never signed in off leave. You're really just documenting everything with the Housing Office to make sure that you only get charged for the appropriate number of days that you were not house hunting.