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.

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.