You can for sure move the center around. I don't do that nearly enough. If you look at my example solve from earlier, GuRoux points out that I could use an E2 instead of a U2+D2, which I should have done.I have a slightly random question about the first block. It's kind of hard to explain, so I'll use an example of what I do.
For the first block, would it work well to place the center with the edge instead of the edge with the center? For instance:
Scramble (off qqtimer): L' U' B2 D2 U' F' R2 D' B' F' R D U' B' D' B' D' U2 B L B' U L' U D2
I would do:
x y' // inspection
Uw' R' Uw2 // 2x2x1 (3)
U' M' U' L U2 L' U L U' L'// LB (10)
I know that I fail at blockbuilding(working on it ), but does the idea of placing the center with the DL edge instead of the DL edge with the center seem at all plausible?
BTW: Here's my solution for your first block with your colors:
x' // Inspection
U' L' U2 // 2x2x1 (3)
D L' U Rw2 U L' U' B // LB (8)