Should i just turn off console or completely shut down the xbox series s evertime after...

After only turn off console can i unplug the cord because fans still running .

You should check the power settings on the console, and set it to power saving mode, that way the fans will switch off after a few minutes, then you can turn off the console at the wall outlet and unplug if you so wish.

The party chat issue should be resolved by updating your controller.

It's normal for your console light to flash for a few seconds after you turn your console off,it's just saving stuff to the cloud before completely shutting down.

Mine did that to me but I just turned off the console by pressing the logo on the xbox and after it shut down unplugged the power brick for 5 seconds the plugged it In and turned it on. Fixed it right away, I wasn't even able to get into my tv settings to
fix it lucky I complete shut down was all it needed

I just set up my console a few minutes ago, and it also took a long time after the update downloaded waiting at the green Xbox One screen. I turned the console off by touching the power sensor on the console, and the console started to shut down with a pulsating
power light. After about a minute, the LED on the power brick turned yellow, when the console completely shut down. I then powered it up with the guide button on the contoller, and after a short wait on the Logo screen, setup continued.

Maybe it's really a case of "have you tried turning it off and on again?"