When to activate my JR pass

Hi,
The best thing to do is to use the Pass for the times of your longest and costliest travel. Since you on your last day want to go from Osaka to Haneda, then that date minus 14 days is the time time to activate it - in other words, activate it to use your Pass from May 12th.
Unfortunately though with the itinerary you have listed, you will be losing money since the cost of the pass way exceeds the regular rail fares for your planned trips. If you want to get your money's worth, you'd need to add some additional trips. But you have the time and there are many good places you can go visit, such as Hiroshima/Miyajima, Nikko, and [Himeji]. And while it doesn't cost much, visiting Nara would be an excellent addition to your trip.

If you would like to check regular rail fares, you can see them on a site like Hyperdia.

Good luck.