Hello.
I’m trying to have both Cubase and (for example) google Chrome on youtube. But each time only one of them is works. Not both at the same time.
Tried many solution … ASIO Multi client, ASIO Flex (with gui), ASIO4ALL, etc.
Nothing worked.
What is your audio interface? You can route system sound to cubase, play audio from another daw like PT or wavelab to cubase control room, play all at the same time and so on. It all depends on which hardware you use.
I play back audio from the OS with my sound card, and use my audio interface for the DAW.
OK, as you have 2 separate audio device, one for OS and one for cubase, there shouldn’t be any interference between them.
What is your audio interface? Which brand and model?
And what happens if you play e.g. youtube, let it play on the sound card, then while it is still playing the sound, start cubase. If it cuts out the youtube sound, at which point it happens?
I have 2 Steinberg audio interfaces, the main one, the UR22 mk2, and the UR12, which I sometimes use on my laptop.
Regarding playing a Youtube video when I start the DAW: I doubt that it would stop the audio, because the DAW will start with the last used driver, which is the ASIO driver of the audio interface in my case. Of course it WILL cut the audio, when the driver is set to the PC’s sound card in the DAW. 
When the DAW uses the ASIO driver, it will all happily play together.
As you’ve tried asio4all and asio flex, asio multi client is still a fuzzy thing. But have you consider taking another route?
How about trying a virtual sound card for system sound. Some of them allows loopback to input, which then can be selected as a generic asio interface input of an asio driver.
Maybe you wanted more casual switch. Then atm release asio background is the only option it helps.