"If I'm not interested in renting my house for a continuous 90 day duration do I need to file and pay for a license every time a receive a new booking request?"
My interpretation is no. You just need to have a valid license for any time in which you are short term renting.
"Or after 90 days I need to file again and let them know exactly how many nights were rented?"
(1) My interpretation is yes, you need to have a valid license for any period in which you do rent out your place for short-term rental. So yes, if you want to rent continuously throughout the year, you need to file every 90 days. That is how I am approaching it. Or, you can take your rental off the short-term market for a period of time (I don't think it has to be 90 days), and not re-license it for the period in which you are not planning to short-term-rent. Then when you start back up, you need to get a license for the new period, which can be up to 90 days.
(2) You do NOT need to report number of nights rented to nola.gov. Airbnb will be reporting that information directly to the City.
I am not sure what is going on with other rental platforms, except I am pretty sure they are not reporting number of rental nights to the City as of right now. I think Vrbo plans to do so in the coming months but they are not ready yet. That is just rumor; I don't know for sure.