What's an ideal itinerary for a few days in Andulasia this July?

I just did this trip, with the exception of Valencia, in the last couple of weeks.

Yes, it will be hot in Andalusia in July.
Some experienced Europe travelers avoid going to Andalusia in July, unless they have school age children and have no choice but to go in the summer. Or they are high school tour groups.

In the last week of April and first few days of May, the afternoon temps approached 90 in Sevilla and Cordoba, with lots of sun and few clouds.

If you're already committed to July, just don't under-estimate the effect this kind of heat and blinding sun will have on your trip enjoyment and physical stamina. There's a good reason why, for centuries, Andalusians have observed the siesta hours of 1:30 to 4pm.

Toledo is an easy day trip from Madrid, 35 minutes by the AVE train.

IMO your list is good except for Valencia, research the geographical distance of Valencia and travel time to get there, if you really want to do it, otherwise omit it from your list.

Travel time to Granada is significant so your 1 night there will not be desirable. The train doesn't go through to Granada from Sevilla (it's a long story).

You could home base in one or more of the Andalusia locations; we actually home based for all 7 days in Sevilla and did all of the Andalusia locations you mention, except Valencia, as day trips. Granada was a long day but is feasible from Sevilla. Or you could home base in Sevilla and Granada and do the others as day trips by train.

Many experienced Europe travelers try to avoid one-night hotel stays.

You don't say how you'll be traveling, trains go everywhere you want to go except the train isn't through to Granada from Sevilla. Driving can be problematical in the urban areas, which are much larger than the historical centers you've probably been reading about.

As I said, I just did this trip so please feel free to follow up with me either here or by Private Message if I can answer other questions.

I love Andalusia, especially Sevilla and Cordoba, but would not go there in the heat of summer. Head to Green Spain, the Basque Country in the north, San Sebastian, etc. Barcelona will be also be cooler.

Basic rule of thumb is that it takes 1/2 a day for every city move. Cordoba has a lot to offer, but with your limited time, may I propose seeing Cordoba as a day trip enroute to Seville and adding a day to Granada? If you want to see the Alhambra in Granada, you need to secure tickets asap. One day in Valencia is hardly worth the effort but it could break up the travel to Barcelona. From Valencia, I suggest the bus to Barcelona.
So perhaps
July 5 early start from Madrid, store luggage in Cordoba visit and sleep in Seville
July 6&7 Seville
July 8&9 Granada
July 10 Valencia
An easier option is to forego Valencia and fly from Granada to Barcelona. Add the extra day to Seville (or Cordoba). Valencia pales in comparison to the other cities in my opinion, but perhaps you have a personal interest in keeping it on the list. Culturally it is closer to Catalan than Andalusia.

I second the others, Valencia is pleasant but not worth rushing through the other cities on your list, and geographically a bit awkward to fit in your route. A second night in Granada or a third in Sevilla would be better.
As for the heat in July... Well, if you're not sure you'll ever make it to Andalucía at another time, then go with hats, water bottles and common sense. The region is worth a little hardship. But if you're able to plan another trip soon, you might want to consider the other, more northerly suggestions.

You don't have enough time to see Andalucia very well, so I'll join the skip-Valencia chorus. It is not remotely in the same category as your other destinations. Plus, it takes at least 6-1/2 hours to get from Granada to Valencia (or a bit less if you want to deal with airports). And one night in Granada would give you not much more than half a day there--perhaps enough to see the Alhambra, but only if you can get a ticket for a convenient entry time. And there are other sights in Granada besides the Alhambra.

Before proceeding much further you should check to see whether you can get Alhambra tickets. The full tickets (including the Nasrid Palaces) sell out months in advance. If the official website is sold out you can try for the slightly more expensive Dobla de Oro ticket. If still no luck, see whether your hotel can supply tickets. After that you're stuck with paying for a tour. If you cannot get Alhambra tickets, I'd scratch Granada from the itinerary this time around--not because it has nothing else to see, but because you could use that time in your other cities if you won't be able to see Granada's #1 attraction.

Seville is a large city with many worthwhile attractions that are somewhat scattered. Two nights there (just a bit more than 1-1/2 days) will mean rushing around in 100+ degree heat. You could use that Valencia night in Seville to good effect.

And however many nights you've planned in Barcelona, I'm betting it won't be enough, so that's another place where the Valencia night would be better used.

There's a 7-1/2 hour train connection (with the first segment in a Renfe-arranged bus that leaves Granada at 12:45 PM and reaches Barcelona at 8:22 PM. Except for the 2-hour bus segment, the trip is on the very fast AVE train. Obviously, flying would be a bit faster.

If you want to avoid a single, long, travel leg, you might consider taking the route Madrid-Cordoba-Granada-Seville-Barcelona. However, that will not shorten your overall travel time significantly (if at all). No matter how you configure the trip, trying to see all 3 major Andalucian cities in the amount of time you have is going to mean a lot of time spent on trains/buses/planes and traveling back and forth between your hotels and airports/train stations. It would be a drag in perfect weather, which you will certainly not have.

If Valencia stays on the itinerary, the train is the fastest mode of transportation to Barcelona at about 3-1/4 hours.

I will join the others in advising you to skip Andalusia in July if you can. We were there in late September and early October in 2017, and the temps were 90 - 95 in Seville. Madrid, Cordoba, and Granada were in the 80's in September. I remember reading trip reports on another travel forum from people who were in Andalusia during the summer months, and they wrote about being drenched in sweat after walking for 10 minutes. If you have made your airline reservations, then sightsee during the morning and evening, and find hotels with pools so you can swim during the afternoon.

I recommend 2 nights for Granada and 3 nights for Seville. We loved Cordoba, so I agree with the 2 nights there. Skip Valencia.

Has anyone mentioned the heat yet!? Anyway, like most people you'll probably survive, just plan long lunches and perhaps a shady break in the afternoon. Its often a fairly dry heat, Barcelona & Valencia can be much more humid and uncomfortable.

With six days (does that mean five nights?), I wouldn't want to visit four cities. I'd suggest Madrid to Sevilla, stay there three nights, with a day-trip to Cordoba, then onto Granada for two nights, then fly to Barcelona. I do like Cordoba and would recommend an overnight usually, but don't think you have the time.

Valencia is very agreeable and I know quite a few people who prefer it to Barcelona, being much less hectic, as a second visit - but I don't think you have time to include it this trip and it's not a first time in Spain priority, I think. If you do decide to include it then it's easy to get from there to Barcelona via the pretty fast EuroMed rail service by Renfe. I'd be more concerned about the time needed to get from Granada to Valencia - I don't think there is any speedy way of doing that.