Why did a background tab open when I hit Refresh?

That is expected, not a bug. Some providers won't let MyQuota read your usage directly from the page you're on, so it briefly opens a hidden tab to check for you, then closes it automatically. No action needed.

Which tabs get opened?

MyQuota only ever opens pages belonging to providers you have explicitly enabled in the settings. It will never open pages for providers you aren't tracking.

Why does it do this?

Because MyQuota has no backend and doesn't ask for your API keys, it acts exactly as if you opened the usage page yourself to check the number. This happens most often for Gemini and ChatGPT Codex, since both need a fresh, signed-in page load to check your usage — that can't happen quietly in the background the way it can for other providers — so opening the tab is the only way to read the data securely from your own browser session.