What is a good 2x2, 3x3, 4x4+, pyraminx, megaminx and square-1 puzzle in terms of great corner cutting and good rotation speed?
For 2x2 its either dayan or witwo v1. They both perform excellently, the difference is only in how the puzzles feel when solving. The dayan is smooth and silent, witwo is a bit crunchy and louder. There are many more like the new MoYu LingPo and the FangShi 2x2 but they still dont have many reviews on them (Fangshi is not really even out yet), you might want to check on those too.
3x3: Pretty much any of the new cubes are fast and can corner cut anything. MoYu WeiLong is very fast, but so is the Gans III, FangShi ShuangRen, Dayan Zhanchi and Maru CX-3. You should watch youtube reviews and decide which you like best. There are more alternatives still like the ShengShou Aurora, Dayan Guhong v2 etc.
4x4. ShengShou 4x4 is much faster than a MoYu Weisu but it needs a lot of modding so that it doesnt lock up and so that it corner cuts. The MoYu WeiSu is a great choice if you're new to modding. The ShengShou 4x4 v5 can be a little bit better if you really nail the mods, where as the WeiSu is pretty great out of the box.
5x5-7x7: ShengShou all the way. It's currently the best brand for big order cubes. For the 7x7 you might want to consider buying the SS mini as the regular shengshou 7x7 is pretty big no matter how big your hands are.
Pyraminx: I dont do pyra but most of the best solvers use a shengshou pyraminx. It is considered to be the best one.
Square-1: I dont know much about this; If I remember right Calvins Sq-1 is the best but you might want to check this elsewhere as I really am not sure at all.
Megaminx: Dayan megaminx is very good as is out of the box. Shengshou megaminx is great too but it requires mods to be awesome. The dayan megaminx with ridges is the one I have and I do not recommend it; the ridges are just in the way. If you decide to go with Dayan you should buy one without ridges in my opinion.
These are my personal opinions mostly so you shouldnt choose based on these; Infact I suggest you start looking at youtube reviews of the ones I mentioned and decide based on that. Especially with the 3x3 its impossible to say what is the best as pretty much all of the cubes introduced lately are atleast good if not excellent.