I cannot find another word for hand, does this mean that 'рука' is the bodypart from the fingertips to the shoulder?
Actually, there's a word кисть.
But it's used only in an anatomical context, when you specify this particular part of a limb, in other cases you should use рука.
Is that exactly the same as the English palm (of the hand) - or does it include fingers too?
The same as palm. I learned the word ладонь listening to this song: https://www.youtube.com/watch?v=Ra0ozaE-oy0
I very strongly recommend everybody to listen to Russian songs, great learning tool :D
Listening to Полина Гагарина's version of this song (which is utterly fantastic btw) was my single motivating reason to learn this beautiful language
Also, wiktionary is very reliable https://en.wiktionary.org/wiki/%D0%BB%D0%B0%D0%B4%D0%BE%D0%BD%D1%8C
It might not be always absolutely detailed, but its declension tables are priceless.
How do you know when people are talking about hands and not arms? (If you're not there) How would you say I got cold feet? or cold hands?
Oh yes! Thanks, I totally forgot about that word. I thought that they use "рука" for the shoulder as well.