I wish I knew how to play new Katarina. Even with maxing Q first now, I typically do poorly because of having to play very safe early games, and I feel like the chance of being able to gank a side lane (typically either dead, or pushed to the enemy tower, and/or lacking reliable CC) is quite low. If you go to a side lane then you can lose half the HP of your mid tower and a ton of XP.
I think I've tried suggested stuff like DFG, deathcap and/or haunting guise, etc., but still seem to only occasionally do OK.