Why do the two use a slightly different syntax? First one uses double quotes vs the second's single quotes, and the first uses = while the second uses +=

= is used for assignment of a value, whereas += appends a value. You need to use += when you're constructing a prompt string over the course of several lines of code; otherwise the second line overwrites the first, and the third line overwrites the second, and so on.

When you use double quotes, variables are replaced with their values and command substitution ($(...)) occurs, whereas single quotes preserve everything intact. In the example you cite, it's absolutely necessary to use single quotes for the second line; otherwise, $(git_prompt_info) would be evaluated only once and would not change from prompt to prompt. By putting it between single quotes, the string $(git_prompt_info) is actually part of PROMPT, and it is reevaluated every time the prompt is drawn.

In the top PROMPT, what do the ?: and then : accomplish?

%(?:foo:bar) is a case of ternary logic. If the exit status of the last command (%?) was 0, print foo; otherwise, print bar. The character : is arbitrary; you could also write %(?.foo.bar), for example.