A.8.1 Is it safe for me to download PuTTY and use it on a public PC?

It depends on whether you trust that PC. If you don't trust the public PC, don't use PuTTY on it, and don't use any other software you plan to type passwords into either. It might be watching your keystrokes, or it might tamper with the PuTTY binary you download. There is no program safe enough that you can run it on an actively malicious PC and get away with typing passwords into it.

If you do trust the PC, then it's probably OK to use PuTTY on it (but if you don't trust the network, then the PuTTY download might be tampered with, so it would be better to carry PuTTY with you on a USB stick).