You can easily take the train from Cordoba to Barcelona, sometimes with a change in Madrid. There are quite a few trains each day that take between 4-1/2 and 4-3/4 hours. I think you'll find that all the trains from Granada and Seville go through Cordoba on their way to Madrid and Barcelona. Ending up in Cordoba makes the exit train trip shorter.
You could fly from Lisbon to Malaga and then travel on to Granada. If you do that, I think you'd do best to travel from there to Seville and then Cordoba, because Cordoba positions you closer to Madrid/Toledo/Barcelona.
It's true that Mudejar architecture is a good part of what makes Toledo special. Only you can determine whether it's best to hold off on Toledo and spend the extra time at one of your other destinations. It's not a tragedy to miss a place (which is a good thing for all of us). Trains between Madrid and Toledo are fast and relatively frequent, but it is extra travel time.
A lot of the most popular attractions in Barcelona are privately owned and open seven days a week. Quite a few are open into the evening, which provides extra sightseeing hours. I don't think you'll have any problem filling your Monday there. Rick's good about summarizing the operating hours of top sights in major cities in his books. That's a good place to start for Barcelona. There are quite a few places in Barcelona for which you should buy a ticket in advance to avoid a potentially very long ticket line and possibly a sell-out. So you're going to need to visit those websites anyway and can check on the closing days at the same time.
The two Andalusian sights for which you'll want to buy tickets in advance are the Alhambra (which sometime sells out months in advance, so pin that one down as soon as tickets go on sale for your date) and the Alcazar in Seville. The Seville Cathedral also suffers from long lines, but you can avoid them by buying a combo ticket at the Iglesia Colegial del Salvador not too far away. You'll find opening days and hours for those and other local sights in Rick's guide book, but such things do change, so you really should hit the individual websites for the latest information.
It warms up early in Andalucía. You'll probably happier if you aren't there in late May. A lot of folks would find that area more comfortable in late April The perfect time will vary from year to year. You can take a look at actual, day-by-day weather statistics for your destinations on the website timeanddate.com. Check out May 2019, May 2018, etc. I see that Seville hit 91 F on May 1 this year, and the high was in the 90s every day from May 22 through May 31. A lot of the monthly-average temperatures floating around seem to be based on a period prior 2010; climate change has had a major impact since then, worsening the extremes.
Granada will probably be cooler than Seville and Cordoba because of its altitude.