The float valve on the side of the brine tank should not float normally. That is to prevent the tank from overflowing in case of a problem, which you appear to have.
Does your clear air check valve housing have water in it, but the ball sinks anyway? I would think that would mean that your ball is bad.
Does the clear air check valve housing have air in it during brine draw? I would suspect an air leak in the hose to the brine tank, or at the brine float. It could be something else I guess.
Yes, you could just stick a plain tube into the tank, which is how mine is set up. However it may be the case that having the float valve saved you from getting brine on the floor due to overflow. So maybe just do it as a test/workaround. I should probably add a brine overflow preventing float valve.
I am not a pro.