How can I check if all of my motorcycle's suspension parts are working correctly, and carrying an even load?
It depends on the bike. Some like mine have one fork for damping and one for rebound. You could tie a tiewrap round the fork so that its pushed up/down depending on your forks to see how much travel you get on each. You can do the same on the rear.
Something like this:
There could be many ways as i think just a laymen style:
1-- This will be hard- Remove the suspensions system from the bike and place them vertically apply same amount of force on top of them using some weight or any other thing and notice how much they contract the one which contract's more is the weaker one so customize it as per need
2-- If it is a coil suspension you can try yo press the coils(keeping them still attached to the bike) from some tool and measure how much force is needed to press the coil and the one which need less force is the weaker one
again i am saying this is just laymen's style i could think cause i don't know any professional way to do it