Can I touch my baby parakeet?

Okay, so I have two parakeets and we got them a Cozy Keet just to chew on and stuff, but a few weeks later we found eggs. We got all excited and stuff, but after a month of them not hatching I checked if they were fertile with a flashlight, nothing. So when the momma parakeet (Goober) gets bored with her old clutch, I just throw them out. But, about 3 weeks ago, I went up to clear out the old clutch, and theres a pink baby bird in there! I freaked out, searching online, finding out what to do if anything happened. Now its got most of its down and its flight feathers are in and growing. Can I touch it so it bonds with me so it likes me when its older? Any idea what it will look like? Goober the mom is yellow and Towley, the dad, is a greyish blue. When will it leave the nest? Is the nest too high up? Its about 6 inches off the bottom of the cage. Any info will be helpful :)