Why is $\frac{z}{f(z)}$ an entire function?

Question

I was reading the answer to the question "Can we prove that every entire injective function is linear" given by @Zarrax; see https://math.stackexchange.com/a/29822/624787.

I had 3 questions while reading this answer:

1. Why can we shift the $f$ to make it $f(0)=0?$
2. Why is $\frac{z}{f(z)}$ entire?
3. Why is $\left|\frac{g(z)-g(z_0)}{(z-z_0)}\right|<M$ for some $M$ for all $z_0?$

Any help will be very, very appreciated! Thank you so much!