You have a wonderful adventure to look forward to !!
Here are my thoughts on a couple of things that are worth considering.
You could depart in the afternoon of 3 and stop over around Carmel giving you a little extra time for all the scenic stops down the coast.1 - 3 - San Francisco
LA has much to offer but with your itinerary you will likely arrive later in the afternoon and have to leave at reasonably early time next morning. I'm not sure it's worth using that time when you will spend a lot of it in traffic. You could stop over in Morro bay/Pismo beach and go through Santa Barbara and Ventura and go across the top of the city to Joshua tree NP.5 - LA (we have no real interest in LA, other then staying the night)
Heading to Lake Havasu from Joshua tree means you could stop to see the bridge and continue to Williams same day.Lake Havasu was mostly a stopover, could be any other place in that region, only thing I found interesting there is London Bridge.
Tough choice, as you could head out through Monument valley to Moab to see Arches NP and then head to Bryce canyon via UT24 and scenic 12 through Escalante. You would need to spend a full day or 2nights but if you skip LA and drive through Lake Havasu you have freed some time. US191/UT24/12 is a fantastic drive ! (Either way you could consider staying just outside the east entrance of the Grand canyon for your 2nd night at Cameron Trading post.8-9 - Grand Canyon
10 - Monument Valley (perhaps going to Page same day)
From Vegas you really want to drive through Death valley and make a few stops. With that in mind you could stop at Lone Pine, maybe check out the Alabama hills and Mobus arch. Great views of Mt Whitney ! Then stop in Lee vining before heading across Tioga pass into Yosemite valley, take your time, it's spectacular up there !18 - Bishop
19 - Bodie
As mentioned in the title, your trip looks great and these are only considerations I suggest you look at. I would get your NP park lodgings sorted asap, or those close by as they are very popular. September is a brilliant time to be travelling this area !
With regards to ther National parks, purchase an annual pass at the first one you visit as it will be cheaper than buying individually. Secondly, if you wish to go to Alcatraz (recommended) I would book tickets in advance using the official ticket office.
Keep coming back if you have more questions, if not please return and share your trip story with us.