I think overall these are some pretty good photos. I, as someone said before, would like to see the girl or the tree in the open space to the right of the first two. Regarding the third, it seems a little blurry, either due to slow shutter speed or missed focus. It also seems a little yellow; maybe try correcting the white balance. The fourth is cute, but the fingers at the bottom are a little distracting. I like the placement of her relative to the string of lights going down the rail. The fifth doesn't really do it for me. However, if you're the parent, I can see why you like it. The sixth is ok, but it'd be nice to see the whole tree.
As far as on camera flash goes, you play the cards you're dealt. Not everyone can have off-camera flash, especially while first learning photography.
As far as cutting off limbs go, well, never say never. It is a general "rule" to try to avoid it. However, there are many, many, many, successful photographs, both famous and not famous, that cut off limbs or parts of heads/faces. When you post photos for critique, especially as a beginner, you will get a few people (who were "beginners" not too long ago) tell you to do this and do that because a book or website said so. We are all beginners to a degree. Photography is an art. There is definitely room for flexibility. However, it is a good idea to learn the rules, then realize when to bend them or even break them.
P.S. Two tips: you may get more c&c with fewer images; you may get more c&c if you number your images.