Yes, sometimes the link is still live, but the tag has changed to nofollow.
If you don't have a subscription, Google Search Console has a free Links report. It's not as detailed, but it shows which sites link to you. (referring domains)
Yeah, anchor text is worth checking too. I do check both.
Right, if the site starts losing traffic, the link is worth less even if it's still live. Better to replace it then.
The spreadsheet idea is good & simple, but it works. I check regularly for my clients with Ahrefs and Semrush, and sometimes manually too. (With the report I provided).
Monthly checks on the links you paid for are good; no need to check every small link daily.
Yeah, that's true; sometimes they remove just the link, and sometimes they delete the whole post. Ahrefs' lost backlinks report can help you find both.