Madrid to Andalucia to Barcelona--best option?

Since you'll be visiting all big cities, I don't think a car is necessary, you can always do day trips via bus and train, or even hire a tour guide for the day with their own wheels.

Spain is a big country and driving will take longer than if you use the high speed train. Also finding parking in city centers can be a hassle, and there could be many places that are pedestrian only so it's easy to get a traffic fine. Finally you'll find petrol is more expensive in Spain than probably where you're from in the US.

I would fly from Granada to Barcelona.

We went right to Toledo on arrival to MAD and spent 2 nights
It was a great place to get over jet lag
Then the 30 min train to Atocha was no big deal- from there we hopped on a fast train to Sevilla
( saved Madrid for end of trip)

We had more time than you -4 nights in Granada was too much and we liked Cordoba a lot more-spent 3 nights there

We did the Madrid(2)-Granada(3)-Sevilla(3 with a side trip to Cordoba) last fall using only trains. No need for a car. You will have less hazzle without a car especially if your room is in the older part of town (pedestrian only zones and narrow streets).
Doing a day trip to Toledo is a good idea since all trains south pass through Madrid. Another option is that you head for Toledo the same day you land in Madrid since all your bags are packekd already and stay 1 or 2 nights in Toledo and head for Madrid the next day. From Madrid you can do Sevilla-Granada or Granada-Sevilla wichever fits you better.
We had been in Madrid before so that is why we only spend 2 nights there but 3 or 4 will be good.

I'd suggest spending the first two nights in Toledo. It's really not a problem to start a trip to Cordoba or Seville with the quick trip from Toledo back to Madrid. There are a lot more trains running along that stretch of track compared to the one into Granada.

Your time is tight, so I can only suggest digging into your guidebook and considering how much you want to see in Cordoba vs. Granada. I was glad to have 3 nights (2 full days) in both those cities for just local attractions. The Alhambra can take up a good chunk of a day if you want to see the entire complex; the Mezquita in Cordoba doesn't require as much time.

I'd look first at the train options into Granada, since there aren't all that many of them. Will you be able to travel at the time of day you generally prefer, or will you have to compromise on the timing?

I'd definitely forget all thoughts of a car. You don't have time for side trips.

Since you’ll be visiting cities the train would be the most sensible option. We have spend two weeks in Andalucia and had Malaga as a base but we wanted also visit the country side and then a car makes more sense. We had no trouble finding a parking space while visiting Cordoba and Granada btw. The Alhambra will take a whole day to visit properly. The Mezquita is absolutely beautiful (been there twice and it still was amazing the second time) but won’t take long to visit. But strolling in the Barrio Juderia and other neighborhoods does. You will enjoy Andalucia!

We're wondering if it would be better to take a day trip from Madrid
to Toledo (which seems like a shame but unavoidable)

Making Toledo a day trip is by no means unavoidable. Yes, it's true that going by rail from Toledo to Andalusia by rail requires changing in Madrid, but it's relatively painless (lots of trains between Toledo and Madrid, and it's at the same station). I would not let this one train change force you to make Toledo a day trip if you don't want it to be!

For your other questions:

Will this second option work better? Can we fly from Granada to Barcelona at the end of our Andalucia portion of the trip? Or would the train be a better option?

Vueling operates nonstop flights (1h30min flying time) and Renfe operates typically 1 nonstop train a day (6h45min rail travel time). Of course, the overall difference is smaller than the travel time difference implies, since flying requires some built-in time for security and travel to/from the airports.

Personally I think I'd be slightly more likely to take the train (in part because I refuse to take cabs to/from airports if at all possible, so that usually adds additional travel time), although 6h+ is at the very top of my personal acceptable rail travel time. I'd look at price and timing (i.e. the train seems to get in rather late) to make your decision.

Once we get to Sevilla should we stick with trains to go to Cordoba and Granada or get a car and return it when we get to Granada? Will a car be an asset or hassle?

If you are going only to Seville, Cordoba and Granada, there is no need for a car (extenuating circumstances aside). Cordoba is halfway between Seville and Granada, so I think the most efficient routing is Madrid > Seville > Cordoba > Granada.

It's true that it's much easier to visit small villages and the pueblos blancos by car, but your trip as outlined is only big cities.