You can test if a command exists in several ways, but the most portable one should be using command -v which is POSIX.

if command -v playerctl &> /dev/null
  # do things
else
  # warn and exit (or prompt for installation)
end

You could always run whereis playerctl and parse the output with awk or sed

That depends on your distro. On Debian/ubuntu(-based) distro's, you can try apt list --installed playerctl, which will output nothing if the package isn't yet installed, or the package name with version info if it is installed. To install it, run sudo apt install playerctl

On arch(-based) distro's, you can try pacman -Qi playerctl, which should return an error if there is no such package installed, or several lines of info if it is installed. To install it, run sudo pacman -S playerctl, ideally after running a full system upgrade with sudo pacman -Syu

If you don’t want a playerctl dependency you can use a dbus library to check for mpris players manually, though it’s a little less readable