I'm curious about the badges involving "unique IP addresses" and what exactly qualifies as a unique IP. I've begun telling my team members to use this site to expand their knowledge and I even refer them to questions I have seen on here for training or troubleshooting purposes. Will that count towards the unique IP total, or does SE see our whole domain as one single public IP?
That depends on the network setup at your company.
You and your team members can try to compare your public IP addresses. You can see your public IP address by googling "what is my ip" and it'll show above the search results.
If they are different then their visits will count as unique, if they are the same, then they aren't unique, so you should ask them to use the links at home. :-)
Also remember that for the visit to count for your badges the link should be from the Share link, below the question/answer not just the pure address from the browser address bar.