What's the difference between "hem" and "hemma"?
The former means home, the latter means at home.
See this thread.
Tack för föreklaringen!
Could nyligen kom vi hem also be used, or is it uncommon?
It's less idiomatic but perfectly fine.
Vad betyder nyss till skillnad från nyligen?
nyss is more recently, a better translation into English would be a short while ago
Tack. Alltså, just now, ungefär?
Ja, det är bra.
What would be the best translation for We just came home? Nyss or nyligen?
nyss, definitely - nyligen would typically mean "recently".
Why is it "kom" and not "kam"?
kam is German, 1/3 person singular - in Swedish, it means "comb". :)
Why is it nyligen? Is this a Plural? Would it be different for "I came home recently"?
No, nyligen is the adverb "recently". It doesn't have singular or plural forms.