Will this mount on a Henry .22 rifle h001?

Laura, this will not fit your Henry Rifle. If I'm not mistaken, your rifle should have a grooved receiver on top for a scope mount. If that is the case, this is the model that you are looking for: Williams Gun Sight FP-AG Receiver Peep Sight with Target Knobs, Black (14170)

What is the distance from the middle of the aperture to the top of the receiver?

That's going to depend on where your mounting holes are on your rifle.

Does this sight have the target knobs for windage and elevation adjustments? The Williams part number indicates that it does, the pictures do not.

The TK is for Target Knobs, and yes the picture is NOT correct. I bought this in Nov/Dec 2019. It does have the target knobs.

What is the distance between the two side mounting screws (center to center)?

I'm not sure I am capable of answering the question precisely enough for a gunsmith to drill and tap receiver properly. What a can tell you is that Winchester lever gun receivers are drilled and tapped accurately to accept that sight.