Answer 1

depends on what you want from this button
if you need only style with some custom functionality or none at all - just go to the website where you saw that, open source code and find how it is implemented.
if you want a button that will actually subscribe/follow your site - there must be some feed channel between your site as feeder and people as readers.
if your site is something like a blog or a news - you might be interested in RSS (for some people who still use it as feeder). the other way is to sync publications with feed in some popular resource, like Instagram, Twitter, Telegram etc. - depending on your audience.