"3. Is there a formula to convert linear axle movement to vertical chain 
deflection? That is: if I measure 1 3/4? of chain sag from the horizontal, 
then pull the axle back so that the chain is now horizontal (I know that 
this term is inexact), is there a formula that will tell me how much the 
axle will move laterally?"


Just study up on your catenary curves, and you'll be covered.  Wikipedia, 
Wolfram MathWorld.  Lots of places.  

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