Open the page that links to you, find the link that points to your domain, and read its rel attribute: no rel value (or one without nofollow, ugc or sponsored) is a followed link. Also check the page for a robots meta tag with nofollow or noindex, which cancels every link on it. A dofollow checker does both in one step from the page URL and your domain.
Every link is an a tag with an href. If it also has rel="nofollow", rel="ugc" or rel="sponsored", search engines treat it as a hint rather than an endorsement. Values like noopener or noreferrer are about browser security and do not change whether the link is followed. Use view source rather than the browser inspector, because view source shows what the server actually sends.
A link can look followed and still pass nothing if the page it sits on is marked nofollow or noindex in its robots meta tag. Directory listings and profile pages sometimes do this, so always check the page as a whole, not only the link.
No. noopener and noreferrer only affect how the browser opens the link; they do not change how search engines treat it.
All three are hints. ugc marks user generated content and sponsored marks paid or affiliate links; nofollow is the general one.
Yes, which is why some links look different in the inspector than in view source. Search engines that render pages may see the final version.