Yeah, you pretty much got it.
All about Catra from the cartoon can be learned here (with pictures!):
Catra is crazy jealous of Adora because Hordak favoured her and gave her the Force Captain position. Catra has tried to take down Hordak a few times and teamed up with Scorpia to do so. She probably wants Etheria for herself. I forget her plans when she took down Hordak.
Catra was the main villain, but Hordak is in the first comic shown kidnapping Adora. She has her own army of green catmen and catbirds (house cats with wings). She's also a very powerful and evil Sorceress. She changes into a cat the same way as on the cartoon, but she becomes an ordinary black house cat (beige when Castaspella turned her into a cat).
Clawdeen was her mentor and taught her how to turn into a cat. I believe she's in love with Bow and wants to force him to be with her (because of the one mini-comic which shows her thoughts for when she makes a fish circus or something. ). It's very much like how Skeletor always wanted to kidnap and marry Teela in old media...and he cloned the Sorceress, which techically makes Teela his daughter. Creepy.
In a U.K. book, I'm pretty sure Catra has her own robots which are like Horde Troopers...but look odd. She also has a cat named Magicat that's basically a white Panthor with red Panthor saddle with 3 jewels instead of the spikes. The same book shows the Fright Zone as a giant cat fortress...so I don't know if that was supposed to be THE Fright Zone or Catra's base on the Fright Zone's property.
She created a wearhouse that made robot clones of She-Ra, Bow, Glimmer and Madame Razz in another book. I didn't fully read it yet, but I think they were supposed to replace them and bring down the rebellion.
Toy Catra is basically Skeletor and Evil-Lyn as one person. In the later minis, she's basically just a REALLY peeved off, jealous girl from high school with a case of the crazies. The "they'll all love me if I get this" and "they won't be able to have the dance without their music! Ahahahaha!" kind of deal. Cartoon and Book Catra are basically Skeletor.