What you most likely have is a tyre with very stiff sidewalls AND a very tight seal on the rim.
The tube is no longer inflated (hence your experience with the valve) and the air intake you are experiencing is a result of the stiff tyre trying to regain its shape and sucking air in at any point where there is not an air tight seal (so likely either the valve hole or valve itself with a punctured tube).
When tyres are a super tight fit on the rim, there is a technique where you can lay your wheel flat on the ground (somewhere soft like grass is best to avoid scratches) and stand on the tyre (not the rim!) with your heel. This should be enough to break the seal with the rim and get tyre levers in to remove the tyre.