For the most part, Carole & Tuesday is a variation on the classic A Star is Born sort of story; you have two amateur musicians from vastly different backgrounds connecting to overcome early adversity and make music that moves the masses. The characters are mostly fun, but the core problem here is that much of the music the show features is pretty generic (particularly the songs in the first half or so) and doesn’t match the characters who are supposed to be singing them. Listened to on their own without the visuals some of the songs are surprisingly good, but almost none of them work within the context of the show.
It also has a number of other things going on as subplots, such as anti-immigration politics and corrupt businessmen, an obsessive stalker, fame-based breakdowns, a conniving manager, parental abandonment, and advanced AI supplanting humanity. All of these end up more engaging than the main story… which is more damning than praiseworthy.
As for the Index Accelerator spinoff, it appears to take place sometime before Index III and mostly features a bunch of new characters who don’t really have much going for them. Accelerator himself is a pretty annoying/one-note character (albeit not as bad as Touma), so its weird anyone would make a series nominally centered around him to begin with. Another oddity is the rather large amount of situational ecchi; girls wandering around and/or fighting in their underwear is quite common.
Speaking of ecchi: Don’t watch Plunderer unless you’re fond of sexual-assault level harassment guised as cartoonish comedy.