1. Can I apply and appear for a Visa renewal *before* expiry of my current F1 visa?

 Yes, you can apply for the new visa before your previous visa expires. Please note that the previous visa will be cancelled or taken up as part of the process of issuing a new visa.

2. If I appear for my Visa interview *before* my current F1 expires and my renewal is rejected - will I still be able to return to the United States on my current
unexpired F1 visa?Or will they somehow "cancel" or "revoke" my existing F1 visa if the renewal is rejected?

It depends on the Consular, he/she may cancel the current Visa as well.

3. Even if my current visa is not "revoked" and is valid, will I still be allowed to enter the country on it? (I understand a visa is not a guarantee of entry).

Yes, if you have a Valid Visa and not "revoked", you maybe able to reenter US.

4. What influence, if any, will a rejection on my F1 renewal have on future visa (and other
immigration related) applications ?

First of all, if you're a continuing student and in good-academic standing. the chance of getting F1 renewal denial is minimal.
For future Visa, it depends on the case/scenario.

Good Luck.