Why wasn't the little girl, Newt, not cocooned or killed by the Alien after she was captured?
It's not clear why would the little girl be just resting with that alien glue in there and not be infested with the embryo inside of her.
Newt was very close to having a Facehugger attach to her, but Ripley came just in time.
How we know this is that the impregnation time of a Facehugger takes several hours or more to completely impregnate a host, so she was not already impregnated with the facehugger already dropped off by the time Ripley arrives.
Once a Facehugger is securely attached to a host, it inserts its proboscis down the victim's throat. This proboscis both supplies the host with suitable atmosphere for breathing and also implants the beginnings of a Xenomorph embryo in their esophagus, a process that takes several hours.
The other issue is that we don't know exactly how long it takes a Facehugger to be mature enough to hatch after the egg is laid by the Queen, we can only assume that it takes some time.
Being that there was only one egg (there may have been more eggs but, from what we are shown, there was only one egg ready to hatch when Ripley arrived) there for Newt in close proximity, it would mean that from the time Newt was captured, the Queen had to lay an egg for her and have the Facehugger mature enough to hatch so as the impregnation process could begin.
As noted in the comments. Two issues are raised in regards to the egg hatching within 30 mins (as that is approximately how long Newt was captive for). It seems that because the Queen was still fastened to her egg laying tube that another alien had to of placed the egg in front of Newt, so we do not know if the egg was just laid or was a mature egg brought to Newts place of capture.
From there if the impregnation process was to take place it would be several hours before it drops off and we are none the wiser as if she was or was not impregnated with the Alien. So, there was just not enough time for all this to have happened. And the fact that Ripley comes just in time to kill what seems to be the only Facehugger around Newt at the time.