How to determine bond length and angle in a conjugated pi-system (polymethine)?
So what are the average bond lengths and bond angles in this molecule
  and how are they calculated?


1,3-butadiene should serve as a reasonable model for your work.



We can find the bond lengths in the literature.  The carbon-carbon double bond length is 1.338 &angst;, typical for a double bond (reference).  The carbon-carbon single bond length is 1.454 &angst;, shorter than expected due to resonance.  All of the carbons are $\ce{sp^2}$ hybridized, which means that all of the bond angles should be ~120°.

Using this information and vector addition you can determine the length of any polymethine.