I am totally new to optics design. I am currently trying to play around with infinity-corrected objective lens and trying to image a slide with a Basler camera. I was able to get my system working with a $160$ mm tube length, $20\times$ finite objective lens and the camera. I was able to get decent magnified slide images with the setup. As a next step, I am trying to reduce the overall height of the optical assembly. I was reading up on infinity objective lenses and tube lenses to help me in this aspect (,,,,).
Here is where I am stuck at and hoping people with some experience in optical design can help out.
Assume that I have selected this particular unit as my $20\times$ objective lens. The technical specs say the focal length is 10mm and working distance is 13mm. Since its an infinity-corrected objective lens, a tube lens is needed to focus the image the camera sensor. So based on the equations mentioned in,
Magnification = Tube Length / Objective Focal Length
So for an effective magnification to be $20\times$ (on the camera sensor), I should use a tube lens with a focal length of $200$ mm. ($200$ mm/$10$ mm = Magnification of $20$). But this is currently making my unit taller than the 160mm tube length design earlier.
Now my query is
If an objective is already selected (any off the shelf objective lens) can I reduce the overall height of the system, keeping the same magnification mentioned in the objective just by editing the tube lens focal length? (I am assuming it's not possible)
For an objective lens, can the objective focal length be changed from the one that is mentioned the manufacturer's spec sheet? (Again I am assuming it's not possible) I have this doubt because I am confused by what the focal length of the infinity objective lens means. Is it a property of the lens or is it the distance between the sample and first lens in the objective?
In infinity lenses, why is Working distance mentioned as higher than the focal length of the lens ? Shouldn't focal length (as the lens is inside) be higher than Working distance (which is the distance between the tip of the casing and the sample surface)? I read  but it wasn't able to clear this doubt.
Is there any way to reduce the height of my overall optical assembly?
PS: I have a Basler camera with a $1/1.2$" sensor (If that information is relevant).
Hoping someone can help out. Thanking you in advance.