Lots of good stuff going on here......
The two things that I see are effecting your stability..... well... they ARE your stability.
1st - you aren't getting anything from your legs. Guys tuck their legs under to get more arch, but you aren't tilting your pelvis enough to get any arch from them. Either the bench is too low, you're too tall, or you're just too inflexible. If you do want to tuck back under, try setting up with a PVC pipe under your back to force an arch for now and drive your heels down to stabilize your body. OR ( and this is what I'd do) just learn to bench with your feet out wide.
2nd - you are loose and wiggly and that's robbing you of pounds on the press.
Get set up tight and get your whole body tight - however you choose to set up. Drive your feet hard - heels down if you bench tucked under. Or, pushing out and away if you set up wide - pushing your butt up towards your shoulders.
When you take the bar out get everything set and tight before you break your elbows. Legs driving, lats flexed, scapulae retracted, and elbows locked. Especially in training get used to spending some time under the weight. Reset between reps and hold everything for a 3 count at the top of each rep.