For the house placement tool every test I've done on it appears to be random. It also seems like the timer is reset or added to it with repeated retries while it is cooling down. Waiting 15 minutes without trying it and trying again seems to work well. Logging out does not remove the timer, it just puts the timer in a stasis. Say the wait is 15 minutes you waited 1 minute logged out for a week, you still have to wait another 14 minutes after you logged in a week later to use the house placement tool.
There shouldn't be a limit on how many times you can put the foot print of a house down besides the random amount of uses on the house placement tool.
There are other timers for dropping multiple houses and replacing one. Say you had one house and placed another by accident. If you want to replace the original house (resize recommended) or place a completely different house there is a 15 or 30 minute timer. The maximum amount of houses you can place, mind you the first placed will start to decay, is two save the bugged resize command which I've been able to get three down.