In logic, you normaly nest quantifiers. Using Mathematica's notation that would be written:

ForAll[y,Exists[x, x^2 + y^2 < 1]]

Or

Exists[x, ForAll[y,x^2 + y^2 < 1]]

These mean different things because the order of quantitifcation matters.

I'm not sure what your statement means however:

Exists[{x}, x^2 + y^2 < 1] && ForAll[y, x^2 + y^2 < 1]

In the first part of the conjunction, y is being treated as a constant, but then it's being bound to a quantifier in the second part of the conjunction. Are they meant to be different symbols? I'm not sure what it would mean if they weren't. I don't think it makes sense as you've written it right now.