I've read before that they were trying to recreate the rivalry between Spock and McCoy. Whether they did that intentionally or not, it fails with Data because he's too innocent to reciprocate, so Pulaski was always punching down. To me, that did make her a bad character.
I don´t really care if they tried to recreate a Spock/McCoy rivalry. Pulaski is a great character by herself. Don´t let your sympathy for Data cloud your judgement. Pulaski is introduced as a character full of prejudice against synthetic humanoids, who overcomes her bias while actually getting to know Data. She learns to see her mistake and ends up fully respecting Data as an equal, sentient being. It is one of the earliest examples of actual character development on the show and much more interesting to watch than two dimensional, over perfect characters.
She's a great actress doing a fantastic job of playing a very unlikable character. Except that she's not unlikable in the way a good unlikable character is. She's not the way OP is feeling about Q.
But it's not just the way she interacts with Data, which does result in character growth for both of them. I'm sure they were going for the adversarial thing but it didn't work, with Data or anyone else.
He's like Murdoc from MacGyver. You hate the character but you can sense that he's going to be defeated eventually and it'll be a fun ride getting there.