The zeta function has infinitely many zeros in $0<\Re{s}<1/2$?

Looking at some other sources, it appears that to be a typo: It should be "$0<\Re s<1$" not "$0<\Re s<1/2$", i.e. the number of zeros in the critical strip.

In fact, no non-trivial zeros of the Riemann Zeta function occur outside the critical strip, so this restriction is superfluous i.e. the formula gives the number of zeroes in $0 <\Im s<T$. It is in this form that Wikipedia and Mathworld state the Riemann-von Mangoldt formula. Technical sources can be found in both links.

Yes, if it were correct, the Riemann hypothesis would be false. It's definitely a typo, however. You can see, for example, here for a fairly detailed proof; suffice to say that nothing can be done to cut the region of validity down effectively, due to the difficult nature of the $\zeta$-function's behaviour in the critical strip.