for us is tricky. "They took everything for us" would imply at least in my mind, that we now have the things, we just didn't do the taking. Then, the translation would be "Vzali všechno za nás". This would be then more accurately translated into "They took everything instead of us", which is I think synonymous wit "for us".
More common than 'za nás' would be 'pro nás'. Both are possible but there is a nuance. ZA implies that we were supposed to do it but they did it instead of us. PRO sounds more like they took something and brought it to us as an unexpected favor.
It doesn't. nám means "from us" (literally it is the dative case so "to us").
No, "from" does not mean "nam". But here "from us" means "nam". Usually "from" would cause Accusative (od koho/čeho), but Czech verb "vzít" is used with Dative (vzít komu/čemu). With some verbs you just need to get accustomed to the case it is used with.