Relative stability of cis and trans cycloalkenes
Usually trans-olefins are more stable than their cis isomers for steric reasons, like you suggested.  However in small and medium size rings this is not the case; here the cis-cycloalkene is more stable than the corresponding  trans isomer.

trans-Cyclooctene is the smallest trans-cycloalkene that is stable at room temperature (trans-cyclohexene and trans-cycloheptene have been detected as short-lived intermediates).  In your drawing up above of trans-cyclodecene (a) draw in the two hydrogens on the trans double bond.  One of them is "inside" the ring and causes steric problems with the carbons on the other side of the ring as pictured below.  Because of this destabilizing transannular interaction trans-cycloalkenes with up to around 11 carbons do not exist as planar molecules.  They exist in a conformation as pictured below for trans-cyclooctene.

 

(image source)

If you can, build a model of trans-cyclooctene.  Notice how you have to pull and stretch the methylene chain in order to attach that last methylene back onto the other end of the double bond - there is a lot of strain in this molecule. Now build a model of cis-cyclooctene; it is easy to loop the methylene chain from one end of the double bond to the other end.  

Not until you get to around trans-cyclododecene (12 carbons) do you have enough carbons to easily connect from one end of the trans double bond back to the other end.  trans-Cycloalkenes with less than 12 carbons have additional strain because of this difficulty in spanning the double bond. Once you get to around 12 carbons there is no difficulty spanning the double bond and so once again trans-cycloalkenes become more stable than their cis counterparts.

Since double bond carbons are $\mathrm{sp^2}$ hybridized with 120° angles, if we add more double bonds, the 120° angles make it more difficult to loop the more flexible $\mathrm{sp^3}$ methylene (with their 109.5° angles) chain around the double bonds.