South Australia, road trip to Alice Springs, flight to Darwin - Car rental? Tips?

When I was planning my Australia trip two years ago I relied heavily on the Trip Advisor Forum for each region. Try posting there, and even perusing the archived topics will be useful. Weather and time of year factor mightily into all the advice. I flew directly to Uluru from Cairns, then back to the US from Uluru through Sydney for my July/August "winter" trip. I did see the start to the road between Alice Springs and Uluru, from the Uluru access! The domestic flights I took were on time and efficient and with distances so vast you might look into comparing costs of renting a car vs flying during this part of your trip, especially if you can't drive long hours.

I strongly suggest that you rethink your proposed March visit to Darwin. That's at the time of the annual monsoon when it rains (in torrents) about 3 days in 4 - not a great time to be visiting anywhere in northern Australia, including Cairns.
Alice and Uluru should be fine that time of year however.
Unless I had unlimited time available I wouldn't drive the route from Adelaide to Uluru. There's a reason it's referred to as the "ghastly blank" - it's featureless desert most of the way. Better (IMO) to fly - even if you have to double back and connect thru Melbourne. Jetstar is the Qantas low-cost airline and has flights from Adelaide to Melbourne for less than AUD$70 if you book far enough in advance. Melbourne has direct flights to both Uluru and Alice. As I recall there are no direct flights from Adelaide to either one unfortunately.
Jetstar in general is very handy for stringing together cost-effective itineraries within Australia. It's a typically spartan low cost airline, but it'll get you where you want to go at very reasonable prices. Do check it out.
If you're open to more "weather friendly" alternatives I can suggest Perth - one of the prettiest and most remote big cities on earth, with terrific beaches and close proximity to Fremantle, which is one of our favorites.
March would be a great time to visit Tasmania too, if you're interested in hiking thru some of the most beautiful national parks in all of Australia.

Thank you both, you have given me a lot to think about.
The first change that I implemented was to decide to not do the drive between Adelaide and Alice Springs. I followed your suggestion, Robert, and decided to fly Jetstar to Ayers Rock/Uluru from Melbourne.
I’ve allocated 3 nights for getting from Adelaide to Melbourne, and 3 nights in Melbourne. It looks like there’s enough to see & do around the Twelve Apostles to spend two nights close by.
Would you have any suggestions of places to stay & stop along this route? Should I stick to the coastal road all the way, or is there a pretty town inland that I should consider visiting?
Unfortunately, one of us has mobility issues now, so hiking is not in our plans. :(

We loved Australia.
I would recommend visiting Melbourne as well as Sydney.

Don't miss the Blue Mountains just west of Sydney.

Also, loved the Great Ocean Highway and Penguin Parade at Phillip Island from Melbourne.

The coastal route will be the more scenic while driving from Adelaide towards Melbourne, especially the Great Ocean Road, which is variously defined as starting from Port Campbell. Along the way you'll pass thru a number of charming little Aussie towns that invite stopping and meandering, particularly if you happen upon a market day or some other special event.
Near the pretty little village of Port Fairy (good place for an overnight stop) is the Tower Hill Wildlife Reserve - one of my wife's favorite places in Australia. Situated in an ancient caldera with a natural moat surrounding it, it has attracted animals from all over the area - you're guaranteed to see Emus (usually with chicks) wandering around, Koalas in the trees (in the parking lot no less) and more grey kangaroos than you can count. There are light duty walking paths there if you so choose, but you can see plenty on the short stroll to the excellent visitor center. We make it a point to stop there whenever we're in Victoria.
The rest of the drive from the Apostles into Melbourne is best enjoyed at a meandering pace - it's just a very pleasant and scenic drive. Stop whenever the spirit moves you - all of the little towns are a treat. Bells Beach might be worth a look if you're interested - it's a famous stop on the international surfing circuit.
Another possibility for a general meander is Ballarat and the Gold Country northwest of Melbourne.
Once in Melbourne a fun excursion is to jump on one of the trams, which the city is famous for, out to the Victoria Market. Makes for a very pleasant day out. You can also check into day trips and tours while you're there rather than driving yourself.
I wouldn't recommend driving in central Melbourne. Trying to master the local rules of the road while dodging the trams is a challenge even for Aussie drivers. Suggest dropping the car on your way in and taking taxi's or trams to get around town.
And good for you in choosing to spend time in Adelaide - an underappreciated little gem of a city. You're going to love it.

Robert, you’re precious!
“I wouldn't recommend driving in central Melbourne. Trying to master the local rules of the road while dodging the trams is a challenge even for Aussie drivers. Suggest dropping the car on your way in and taking taxi's or trams to get around town.”

If we were not to stay in the CBD, where would you suggest for a budget-friendly option, with very good public transit access? (One of us will be using a lightweight travel mobility scooter.)

I was just juggling with this problem... where would you suggest that we drop off the rental car? Pricing the car, I may be renting from Budget or Avis. Right now, I have a cancellable accommodation reserved near Flinders St Station. To get into Melbourne, should I drop it off in the CBD, a suburb, town on the way in + train, or the airport??

“And good for you in choosing to spend time in Adelaide - an underappreciated little gem of a city. You're going to love it.”

With the re-shuffling, I was able to add a night to Adelaide. :)
Thanks!

One more question:
If we were to stay between Lorne and Queenscliff, is there a way that we could get to Phillip Island, that would bypass Melbourne? The map seems to indicate a ferry to Sorrento.

We will definitely be wanting to see the penguins on Phillip Island, and doing it on our own rather than with a group may make more sense to us. We would likely overnight in San Remo before dropping off the car the next morning, if we were to do this.
Thanks again!

Tom, thank you.
Yes, we have reworked our itinerary to eliminate the drive to Ayers Rock/Uluru.
Darwin, despite the timing, is still on the itinerary for personal reasons. Given the season, I don’t think that we’ll get the chance to squeeze an overnight to Kakadu, unfortunately.

Regarding accommodations: we've had good luck using booking.com on our own trips. Their filters are quite handy for sorting thru the various options and the map feature is helpful for identifying places close to public transportation. I see that their Melbourne city map is quite detailed regarding the bus and tram routes.
The Melbourne "hook turn" is famous within Australia, and I've seen Aussie's from other parts of the country shudder at the mere mention of it, though it becomes pretty intuitive once you do it once or twice. Still, it might be a good idea to google it to see what you'd be in for if you stray into the CBD.
As I recall there are some toll areas downtown that you need to pay for within 24 hrs. (by phone) to avoid fines. Maybe Mopak has more local knowledge about those. I don't recall them being particularly expensive or onerous - just something to be aware of.

Also, if you're still committed to Darwin, know that there are direct flights there from Alice Springs, while it's a bit more complicated (and more expensive) to fly there from Uluru. I'd also suggest booking the earliest flight out of Alice so as to arrive in Darwin by early afternoon. Some of the late afternoon thunderstorms that go rolling thru there are truly awe inspiring - not a good time to be trying to land...
Be aware too that the drive from Uluru to Alice takes about 6 hours. Many people mistakenly believe that they're virtually co-located for some reason. Along the way you'll probably see herds of wild camels wandering thru the bush - the descendants of those used to build the Ghan railway years ago. Once the work was completed the camels were released into the wild where they've thrived. Interesting factoid: there are more camels in Australia than there are in Saudi Arabia ... or any country in the Middle East for that matter.