Can You Half Infinity?

It's infinite. One way to look at it is to realize that if you added two finite things together, the answer is finite, so 1/2 of infinity cannot be finite, hence infinite.

What is the half of infinite?

Since infinity is not a finite concept, half of infinity is still infinity. Infinity over infinity is an indeterminate form.

Can we divide infinity?

The reason that in the usual extension of the real numbers by "infinity" and "minus infinity" you cannot divide one infinite quantity by another has nothing to do with different sizes of infinity. Rather, it is the same as the reason why you cannot divide zero by zero.