Is this a bug or is it supposed to be like this? I just got the Rockstar Pass and I'm playing the DLC traffic case - so far I've gotten no exp for my interviews. I'm also not getting any callups on the radio while I drive for street crimes, is that normal?

Unless you're playing the campaign from the start, I don't think you get XP for any case when you go via the desks. I have not gotten any xp when I replay my cases for higher ranks.

Well that's a bummer. How can I keep ranking up, then? Do street crimes?

Street crimes when doing free roam, find cars and film reels (and the cop badges if you have the dlc).