Net worth includes all items and possessions that have value minus liabilities. Yes, this includes your 14 year old cat if you can sell it for a profit. If you decide to exclude items because they are not currently generating income, then that is fine, but you are no longer talking about net worth. It is understandable if you do not want to include items such as said cat value because it is a small amount, but usually a home value is a significant portion of one's net worth.
Bingo.
It absolutely should be counted.
No, it often isn't counted as part of your FIRE stache, because it's not part of your investment portfolio (and perhaps should/shouldn't be counted, depending on your draw down strategy). That's a separate question. But it's definitely part of your net worth. Whether you intend to sell--or not--is not relevant. Whether you have a mortgage--or not--is irrelevant (aside from counting the mortgage, if you do have one, against the value). It's an asset (and possibly netting to a liability after the mortgage if you're underwater). How you count it is up to you (some include selling costs, some don't; some reduce the value to what they'd get if they had to sell immediately, some don't). But it's definitely an asset.
If you are willing to sell it, count it. If you are never going to sell it, then don't count it.
Like I said above, irrelevant. To illustrate: should I not count any stocks I'm holding that I plan to never sell?
If I have a paid off house worth 1MM, and a stock portfolio worth 1MM kicking off 3% dividends (and like the MMM family, spend about 25k annually without housing costs, so that 30k from dividends covers my spending), and I never intend to sell either, is my net worth $0, because I'm not willing to sell any of it?
Of course not. Net worth is 2MM. That's what I'm worth (on paper). FIRE portfolio is 1MM.
Naturally some don't include it in their FIRE portfolio. But to not count it in your net worth means you are no longer calculating net worth, but some other number. That's fine, but don't try to convince us that what you're calculating is net worth. It's not. :)