I lost my iPhone and i had a lot of important contacts and documents in my iCloud mail account. The only "computer" I currently have available is my friend's iPhone which is connected to his own iCloud account. Is there a way to log into icloud.com from an iPhone?
I'm not exactly answering your question, but if you're looking to change your iCloud password then… If you remember your Apple ID and password, you can login from another non-iPhone device using a current web browser. The address is https://www.icloud.com.
Use Puffin. There is a free version in the app store (you don't need the paid version for this). https://itunes.apple.com/us/app/puffin-web-browser-free/id472937654?mt=8
I use Puffin to (successfully) get around the limitations of Safari/mobile/touch browsers in a lot of difficult situations.
I have just tried, and you can successfully log in to iCloud.com on iOS with Puffin and access Mail. A hint: You may need to pinch-to-zoom out sometimes to get the interface to redraw to fit your screen, even though it's not obvious this will work. Just try it if things aren't displaying right ;)
I don't use iCloud mail, so I couldn't actually try to load a message or read it, but at least I know you can log in.