You're right that your core is being worked when you do compound lifts, but a little bit of specialized core work is always a good thing. I'm a fan of lower rep, weighted core exercises, and I know that BTB loves his weighted situps.
It's good that you're ready to leave your ego at the door. We all started somewhere, and we got to where we are now slowly, by starting with weights we could handle and consistently progressing, always adding weight to the bar. In 6 months you'll look back and laugh at the weights you're using now and how easy those weights have become. That's one of the joys of slow and steady progression.
Don't worry about power cleans. Done improperly they can do more harm than good. You can get the benefits of cleans with other movements like high pulls, which is basically the first 2/3 of a power clean.
One thing I want to mention is that short intense sessions are GREAT. I never train for more than an hour, and BTB's sessions usually run just over an hour. In that hour we bust our asses. When I leave the gym I look and feel like I just sprinted a marathon, usually dripping with sweat. Sometimes I have to sit down for 10 minutes to rest before I can even drive home. If you go in the gym and kill it, then 4 sessions a week at around 1 hour each will be PLENTY to make you grow and progress.
If you can survive 3 hours in the gym then you're resting too long and lifting too light.
Current PRs at 242- 495, 345, 585, 1425