No, they do not fully lock out. I do not know of a suspension fork that doesn't have at least a little movement when locked out.

Higher-end forks are pretty much zero play to me. My 2013 Fox 32 is solid (if it moves at all, it's literally undetectable). My daughter's 2018 Rock Shox SID is even more so--rock solid. Even my wife's budget-priced Rock Shox XC30 is pretty solid--but compared to the other two, it maybe moves a 1/16"--nothing that would be detrimental to one's sprinting or climbing. So yes, they do fully lock out--not all, but most and if they don't it's minimal movement.

Is it a remote version (not sure if available on Judy)? If so, tightening the cable tension will help the knob pull more thus putting it more into lock-out mode otherwise it will bounce some. If it's a manual turn knob, you can't do anything about the play. Many lower-end forks don't achieve the rigidity of higher-end forks (see my earlier post)

They vary a lot. Usually they need a little bit of compressibility to protect the hydraulic damper, and or a blow off valve. The damper (which is what holds the lockout) is generally not designed to withstand the forces of a hard hit while locked out. Also, for many riders having a useful climbing mode that reduces pedal bob while still allowing helpful suspension movement to maintain ground traction is more valuable than a rock hard lockout, and so many trail oriented forks have an intentionally softer climb mode. If the lockout becomes softer than it originally was that's a sign that the fork's damper requires service.