The question is a little confusing. There are two different devices in your toilet tank. The fluidmaster 400 is a (1) tank filler valve with a float that cuts off the water inflow, it does not incorporate a flapper (performs the same function as a 'disc'). Device (2) in your tank is a flushing device, which incorporates a flapper or disc. You do not have to match a flapper/disc with a particular filler design. The two separate devices can even be from different manufacturers. The hole in the center of the disc fits many toilet 'flapper' designs of the 'pecking chicken' type, these bob up and down, they have a have tube that fills with water. The water weight in the tube becomes a counterbalance that holds the disc open momentarily when you pull on the chain. The pecking chicken' type of disc in the picture will not work with a system that use a flexible plastic hanger harness, the 'hanger type' hangs on posts molded into the sides of the fill tube. In the hangar type there is no tube full of water, there is no center flapper hole, it has rubber straps. If you have the 'pecking chicken' type mechanism just measure the diameter of the fitting entering the center hole. Buy the disc, press it on, if it is snug, it fits. No tools needed. The little 'stiffening ribs' in the picture may be a problem. If you install the disc in the picture listen hard for a 'whispering' leak when it is closed. If it leaks, the disc is too stiff, if so, look for a softer disc without the stiffening ribs in the middle, and made of a softer and thinner rubber.
It seems to fit all of my American Standard commodes.
Yes it can be screwed on or pushed on