Hi Max. Welcome. It sounds like you've gone through some of the materials within the "Learn" section here so that's a good start, so on to your questions...
- The competitive niche keywords represent where you want your site to be when it's as powerful as your competitors. No site is going to have a lasting presence in the search engines that finds some loop hole to exploit and suddenly pops up near the top with thin content and zero engagement, so think of the competitive core keywords also as a guide to finding other sites where you're not just getting a link, but getting exposure for your brand or business. For example, a sub-reddit could drive tons of traffic / sales to your site if you're engaged in the community there. Sites with active readership that are writing regularly about your niche are likewise important. You get the idea. For niche competitiveness look at the whole picture.
- For keywords, don't bog yourself down. You need to develop your site and content in a way that would make sense for someone searching for that keyword. Put it together from the searchers perspective for the given keyword(s). What do they find if they're researching or shopping for something they've put into search? Does your site solve their problem quickly and easily? Does it give them confidence in their purchase? Etc. Closely related keywords and derivatives can all target one page, while the amount of keywords you research should be limited to how much content your site contains.
- See search ranking factors for an extensive breakdown of rankings: http://moz.com/search-ranking-factors. Search and inbound marketing is a huge business. You'll be will on your way though if you're highly focused on making a site and product people use and love (and link to naturally) than trying to learn every aspect of search.
- Moz shows the page that ranks in your selected search engine as the page that is targeting that keyword. So if your keyword doesn't rank, there's no page associated.
- Use that data in Moz to help you find even more connections than you would on your own. Remember the Moz tools are like your sleepless robots that are collecting way more information than any one person ever could just working on their own.
- As much as possible, don't think of your future relationships with other websites as 'link partners' instead think about how the aspects of your business mesh well with what they're doing. If they write an entire well-read article saying they love your business but the link is just to the homepage and a nofollow at that, it still should be considered an outstanding link.
Connect with people in your niche that aren't competitors and put your passion into your business and relationships with them and the links and rankings will come with it. Good luck with your venture!