As for business cards, yes, if they have some non-obvious reading they will have kana or romaji there somewhere. (Usage of romaji on business cards is wider than you might expect, since it's seen as kind of the equivalent of a modern sans-serif logo for a business in some circles)
e.g.
Card with furigana: https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...
Card with kana elsewhere: https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...
Card with romaji: https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...
On the first example -- what is the significance of the character which is a circle within a circle? I'm assuming it's some sort of graphic or punctuation, rather than a kanji...?
The heading of that corner is just the pronounication of "m-take design" written out in katakana. The first line under the heading talks about the type of products the person works on (direct mail, leaflets, pamphlets). The second line talks about their specialities, as in the industries they focus on (cosmetics, health food), and then the last mentions they'll also do logo design, homepage design etc. I guess they wanted to emphasize the second line.
You'll sometimes see kanji-sized single circles used in Japan to indicate omitted characters (and you might see them on business card _templates_ as a sort of lorem ipsum, but unlikely on actual business cards), but this double circle doesn't have any specific meaning as far as I know.
I'm thinking.. I've only ever seen someone's name, will I make serious faux pas if I mispronounce it with the regular reading when I meet them?
(this is less pronounced in English because most of the time you can tell how to pronounce it by reading it. My last name suffers from the fact there is multiple possible pronunciations, so whenever I meet someone who has never heard my name they always stumble and look at me for help)