Chrome extension: Checking if content script has been injected or not

is this safe enough to naively call chrome.tabs.executeScript every
time the extension icon got clicked? In other words, is this
idempotent?

1. Yes, unless your content script modifies the page’s DOM AND the extension is reloaded (either by reloading it via the settings page, via an update, etc.). In this scenario, your old content script will no longer run in the extension’s context, so it cannot use extension APIs, nor communicate directly with your extension.

is there a similar method for chrome.tabs.insertCSS?

2. No, there is no kind of inclusion guard for chrome.tabs.insertCSS. But inserting the same stylesheet again does not change the appearance of the page because all rules have the same CSS specificity, and the last stylesheet takes precedence in this case. But if the stylesheet is tightly coupled with your extension, then you can simply inject the script using executeScript, check whether it was injected for the first time, and if so, insert the stylesheet (see below for an example).

any better method for background script to check the inject status of
content script, so I can just prevent calling
chrome.tabs.executeScript every time when user clicked the icon?

3. You could send a message to the tab (chrome.tabs.sendMessage), and if you don’t get a reply, assume that there was no content script in the tab and insert the content script.

Code sample for 2

In your popup / background script:

chrome.tabs.executeScript(tabId, {
    file: 'contentscript.js',
}, function(results) {
    if (chrome.runtime.lastError || !results || !results.length) {
        return;  // Permission error, tab closed, etc.
    }
    if (results[0] !== true) {
        // Not already inserted before, do your thing, e.g. add your CSS:
        chrome.tabs.insertCSS(tabId, { file: 'yourstylesheet.css' });
    }
});

With contentScript.js you have two solutions:

1. Using windows directly: not recommended, cause everyone can change that variables and Is there a spec that the id of elements should be made global variable?
2. Using chrome.storage API: That you can share with other windows the state of the contentScript ( you can see as downside, which is not downside at all, is that you need to request permissions on the Manifest.json. But this is ok, because is the proper way to go.

Option 1: contentscript.js:

// Wrapping in a function to not leak/modify variables if the script
// was already inserted before.
(function() {
    if (window.hasRun === true)
        return true;  // Will ultimately be passed back to executeScript
    window.hasRun = true;
    // rest of code ... 
    // No return value here, so the return value is "undefined" (without quotes).
})(); // <-- Invoke function. The return value is passed back to executeScript

Note, it’s important to check window.hasRun for the value explicitly (true in the example above), otherwise it can be an auto-created global variable for a DOM element with id="hasRun" attribute, see Is there a spec that the id of elements should be made global variable?

Option 2: contentscript.js (using chrome.storage.sync you could use chrome.storage.local as well)

    // Wrapping in a function to not leak/modify variables if the script
    // was already inserted before.
    (chrome.storage.sync.get(['hasRun'], (hasRun)=>{
      const updatedHasRun = checkHasRun(hasRun); // returns boolean
      chrome.storage.sync.set({'hasRun': updatedHasRun});
    ))()

function checkHasRun(hasRun) {
        if (hasRun === true)
            return true;  // Will ultimately be passed back to executeScript
        hasRun = true;
        // rest of code ... 
        // No return value here, so the return value is "undefined" (without quotes).
    }; // <-- Invoke function. The return value is passed back to executeScript