"Wir haben die erste!"

[dnovinc]

Would any of these be correct "We are having the first!" or "We are having the first one!" ??

[rsalden]

German doesn't distinguish between "We have" and "We are having", so "We're having the first" would be perfectly fine. I would say "We're having the first one" is a better sentence, although not a literal translation. In German it's either "die erste" (the first whatever - context will tell you what) or "die erste Katze", "das erste Mädchen", "der erste Mann" etc.

[duoderSie]

Shouldn't "erste" be capitalised?

[rsalden]

Sometimes it is, sometimes it isn't. Depends on what you want to say and really hard to explain for a non-native speaker. I personally think this sentence isn't right for this level... Reported it to duolingo.