Is it stated anywhere what happens to animals when they die? I think it is stated somewhere they have no share in the world to come, so what happens to them after they die? This topic interests me greatly.
You ask two different questions:
Will there be animals in the WTC?
Yes, here's a simple proof.
If by the WTC you mean the world after the 3rd Temple is rebuilt - of course, there will be animals and everything else as the Gemmorah in Shabbes 65 "דאמר שמואל אין בין העולם הזה לימות המשיח אלא שעבוד גלויות" (Rambam also rules that at the end of Melachim). Meaning there will be no perceivable difference between this world and the WTC, therefore the animals (DanF also remembered the sacrifices!)
It is commonly accepted, that the very idea of our 6000 years world is to fix all the primordial sins (Adam's and the Golden Calf and some more) and to bring the world to the level of "Adam before he sinned".
Since there were animals in the original Eden (see Bereishis), we can assume there will be animals in the last Eden.
Animal soul reincarnation
The details on reincarnation, however, are very scarce, I mean, will you meet the soul of your passed dog or just some other dog - I didn't find that information.