Floor plan with all dimensions would be in this case.

Its 10 ft 5 inches long and 7 ft wide

Here are examples of floating vanities. I like it when they extend from wall to wall for the added support. I also like soft lighting beneath the vanity to illuminate the floor.

so are you planning to build a wall where that 2nd window is, to bump up against the vanity? I'm trying to imagine where this wall would be.

No wall, vanity would just stop few inches before the glass shower. And touching the wall on the opposite side

oh i see what you mean. I'd center the vanity, as long as you have enough inches on either side to clean in between.

as for LED mirror -i have one and no overhead/sconces becuase my bathroom is smaller than yours and it's bright enough (I have an overhead light built into the fan as well). If i were doing an LED mirror in a bigger bath i'd probably also put in some recessed can lights.

You will need good overall lighting in thta space so ceiling pots for sure and no light on the wall with the LED mirro but it is not enough light by itself.

"Does the floating vanity need to be pushed up against the side wall?"

No. The vanity cabinet hangs off the rear wall studs. Side support is unnecessary.

But will it look skewed with a 4-5 inch gap on the side?

why not center it so that there's 2 inches on either side rather than 4 inch gap on one side?