Wrt to 'pushing through the floor', that's how I was taught to deadlift. If you're pulling you're not maximizing your potential, plus, you're also not engaging everything. I also find this is easiest done in socks. When pushing, everything is engaged from the ground up. Everything is tight and feels like a compressed spring. Once core and shoulders are set, drive through your feet as though you are trying to push your feet through the floor. It sounds obvious, but it wasn't until I trained at a powerlifting gym that I realized how much I had been focusing on pulling, and not pushing. And, the difference is huge. To further reinforce the technique I stop at the bottom of every rep, reset, and re-engage. When you're doing 8+ rep sets it's brutal. No momentum at all, each rep feels like a micro set. I like it b/c it empahasizes resetting/re-engaging everything, especially the core. As one powerlifter once told me, if your core isn't blasted after deads/legs then you're doing it wrong. I don't do this every workout, but I do it for one of my two deadlifting sessions each week.
Probably the most noticeable benefit from lifting, esp since I train legs (in one way or another) 3x week, is my core strength.