Please feel free to keep it alive.
Well, Method 2 would give you a false feeling of security, since if your computer is compromised by a keylogger, there is no reason not to expect it to take mouse click snapshots besides keyboard events.
While using Method 3, you could use iptables to prevent your secure user from accessing the network. This can not protect you from privileged keyloggers and such, but could be used to make sure offline-only tools don't do some fishy stuff in the background. It could even be switched on and off manually, to be used in conjunction with tools like blockchain.info/wallet. I don't know if offline tools will be of any use in the case of Ripple though.
I agree that none of these three methods provide absolute security.
1) With the first method you need to trust the Live CD image provider that it's free from malware and security holes.
Open source makes it easier to monitor, but I think there might be a few loopholes when the software is built and the image is created.
I need to look closer into this. Do you know if Ubuntu/Mint Live CD image creation process is completely transparent?
2) With the second method, yes some advanced loggers would likely capture everything you interact with including sceenshots, mouse clicks and moves. You can probably make their life harder if you scroll/resize the page a few times while you type your password with injections.
They would need to capture your screen several times to determine the offsets of injections within your password properly.
3) Yes if keylogger has root access then the third method doesn't really bring much security to online operations. So users should be careful when installing random software from the Internet. It might be wise to keep a small partition on your hard drive with some trusted and well tested clean Linux installation for just the purposes of dealing with sensitive data online. Can you recommend a good one for that?
The only rock solid secure method is to keep your secret/private keys on the protected offline computer and transfer signed transactions via clean USB flash drive or QRcode printer/reader. But that luxury is only available for BItcoin with Armory at the moment, so Ripple users are left with what we have.