Ghost ships with a dedicated Code Injection field built for exactly this — pasting a script into the site header without touching theme files, available the same way whether you're self-hosting or on Ghost(Pro).
Yes — Site Header/Footer code injection is a core Ghost feature available on all supported plans, whether self-hosted or on Ghost(Pro).
Yes — the script loads in the page header regardless of whether the underlying post is public or gated behind a Ghost membership tier, as long as the page itself renders.
No — newer Ghost versions include basic first-party analytics, but that runs independently of any script added via Code Injection, so the two don't interfere.
Yes — Site Header injection happens at the {{ghost_head}} handlebars helper, which every standard Ghost theme includes, so it applies regardless of which theme you're running.