If the debt-to-equity ratio of a company is calculated by dividing the liabilities of a company by the shareholder equity of the company, does this mean you would divide $10,000 (if a company has this much debt) by $15,000 (the total value of 15,000 shares owned by the public at $1 per share) and thus get .66?
not exactly. two things:
there are several types of liabilities, including debt (which is a subset). you should avoid conflating debt with liabilities, as things like accounts payable are also short-term liabilities, and yet not debt
the metric you're describing is max market cap, not shareholder equity.
What happens if there are 30,000 shares that weren't offered to the public?
Would the shareholder equity then be $45,000 and the debt-to-equity ratio be .22?
Insider-held shares contribute to market capitalization as far as I know. Total outstanding shares is published in every quarter's 10q and the 10k. So if shares exist that are not owned by the public, but rather held by insiders, it doesn't matter. The calculations for market cap remain the same. In the event that a company buys back shares leading to dissolution, the number of shares outstanding decreases, but the price per share often increases to compensate, so there may be no substantial variation in market cap.
None of this affects shareholder equity, though.
What exactly constitutes shareholder equity?
Assets = Liabilities + Equity
Shareholder equity is also (or very nearly equal) called "Book value", used to calculate BVPS and P/B.
This video shows a graphic; the gray area is, well, guess what it is.
You can also get a more in-depth explanation of how to read balance sheets by various YouTubers:
Financial Education, https://www.youtube.com/watch?v=rhHt15F55AA