So I took a look at your file and noted these things:
Most of your problems now and future ones you are yet to find are associated with this below:
You have an unapplied scale on your armature in Object Mode - this you remove by selecting the armature in Object Mode and key CTRL+A => “Rotation and Scale”. also your armature and mesh have an unapplied location, put this right in the same way but chose “Location” after CTRL+A. Always do ALL edits in EDIT mode, including overall scaling, rotation, etc. If you must scale in Object mode always apply the scale before you try to parent the mesh to the armature. :ba:
The driver not resetting to 0 was down to a little error in the driver/shapekey setup, I reset the Transform type to “World” then back to “Local” to correct it - I don’t really know how it got offset, but this tip corrects these type of things!
Your bones used to control the movement of your shapekeys are not restricted to their target axis, so if moving a bone along its Y axis is all that is required to activate the shapekey, lock the other two axes on the bone as below:
That makes it easier to understand your rig for an animator, including yourself if you are the animator!
The weights for the parent need adjusting, you have an issue at the knees of the back legs, but this you can sort later. some vertices are folded inside the leg. If you have not tried Weight Painting yet, do it slowly and a bit at a time, keeping backups of your file at each stage in case you mess it up, then your can go back to a good version.
The eyes need some work as when you activate the “Blink” function, the eyelid goes onside the eyeball, I suggest you either use a different method to close the eye, like rotating a bone rather than a shapekey or slightly reshape the eye itself.
I have added a Sub-division modifier to your mesh to make the display look smoother - you should read up on Sub-division mods in your spare time… Note I put the Sub-div modifier below the armature modifier - that makes the file more efficient and is good practice.
I like the idea and I like the work you have done so far - keep up the good work and ask here if you have any questions. It is easier to use scripted expression to drive shapekeys, but the way you have done it is good for your level of expertise. You could look at scripted expression at some stage, but not just now.
Anyway - here is your blend file back with my mods, including the bone location locks and some other bits and pieces.
frogs charachter 2 .blend (1.56 MB)