What should I do if pop-up window is not showing up after clicking Checkout button in pop-up cart (dropdown cart and sidecart as well)?

Our app works only on standard cart page.

If you want our app to work for you, then you need to go to your Shopify admin panel, there go to Online Store -> Themes -> Customize your theme and there (maybe in Cart page section, but it can't be said for sure because all themes are different) you need to switch your cart to "Page" mode or disable ajaxify.