HOMO Odd, LUMO Even: Why does the affected orbital change depending on the carbon? How are carbons numbered?
To start with, I'll address the frontier orbitals. Lets look at what the HOMO and LUMO look like.



Notice the LUMO has no density on carbons $1$ or $3$ and (while it is a little difficult to see in this image) little density on carbons $5$ and $7$. Similarly the HOMO has little to no density on any of the even carbons. So, if we want to affect the energy of these orbitals, we have to add a substituent to a carbon that has electron density from that orbital. The odd-trend is likely not a general rule, but if you know the HOMO and LUMO reside mainly on different carbons, you can assume that selective functionalization of those carbons will affect the orbital of interest. I say "assume" because you should always keep in mind that the orbital picture is an approximation and that once you make a substitution you have to check to make sure you haven't perturbed the system enough where your new compound doesn't have frontier orbitals that look like this anymore. 

In regards to the numbering, this is really just a matter of convenience for a fused ring system like this. We don't number these bridging carbons because they can't be functionalized; that is, we can't add a group to these carbons without turning the molecule into something other than azulene. So the numbering starts from next to these bridging carbons and skips over them as you move around the rings.