I just started, and I was going through Trollsblight caverns, just quickly running through. I accidentally tried to steal the eggling from the sylvorpa, and then the battle triggered. I tried to catch it, it ran, and I went to my hatchery to see the eggling. But, it wasn't there? Did something happen to it, or was I not able to obtain the eggling in that event?

In those early zones, occasionally the eggs can break and die. :\ It's pretty sad, but most likely that's what happened. Next time you see an egg, be sure to look at the explore/battle log area. It should tell you if you successfully get the egg or not.