4. Self lubricating cube.
I actually met someone who had already done that at a competition. He had one of the old type A cubes, and he took one of the corner caps out and replaced it with a small block of sponge soaked in silicone lubricant. It functioned perfectly fine, and he hadn't lubed his cube in about 3 months.
And for my own answers...
1. Smooth and gummy. I tend to lose control if it's too glidy. I've always used cubes that were a little tougher to turn than the better ones out there. Right now, I use a Black Lubix Lunhui that has a VERY gummy feel to it, and that cube has served me much better than my rather slippery White Zhanchi. It also locks up less than my Zhanchi. Maybe if I got a Lubix Zhanchi, my opinion of the Zhanchi might change, but for now, the Lubix Lunhui will be my main.
The regular Lunhui is also quite nice with Maru lube. Even though it's very glidy, it's still very controllable due to having a bit more of a blocky feel. The corner cutting is the same as the other Dayan cubes, but it takes slightly more force to cut corners, if that makes any sense.
2. I use a lot of corner cutting because my turning accuracy is absolutely terrible. But I don't think I would need more corner cutting. Is it even possible to add more corner cutting without ruining the basic internal structure of the cube?
3. Lookahead for sure, and also good solutions. My TPS is crap, especially because I don't use my left hand enough.
4. Less lock-ups will never hurt, but for now, I don't think improvements will be necessary.