Yes, they should retain their size. But if I start on straight needles, then switch to circulars as the project increases in size, sometimesI notice a difference in the way it looks. Probably has to do with a silght change in the angle of the needles.
I don’t find it easier or harder to use smaller versus larger.
It’s the needle tips that determine the size of the stitches, the cord just holds them. However, if you knit very tightly, that would stretch the yarn, so when they rest on the cable it would relax and you might have a hard time getting them back up to the other end when you need to knit them.