In the GEE code editor, I can only activate the 'save' button, while the 'run' button and other buttons always keep gray on my computer. I tried my same account on another computer, it works well. Is there any solution to solve this problem?
This means there was a problem initializing the Earth Engine API client. Usually you should see an error message pop up, but if it hasn't yet timed out, or if you dismiss the dialog, you will get this state with grayed-out buttons (and no documentation).
Reload the page. If that doesn't help, you may have some kind of browser extension/configuration interfering with loading, or a problem with your account, and you'll need to troubleshoot that, possibly with the help of someone who knows enough web dev to see what your browser's dev tools/console say about what's happening.