I have some question about the usage of "see":
- He noticed that she was smiling.
- He saw that she was smiling.
Could the second sentence be uncommon. Would the following:
- He saw her smiling.
be more common?
#2 is not uncommon, and I don't think one could say with any certainty that #3 appears more frequently than #2.
I think all of your examples are normally used.
However, "noticed" has more of a nuance of "caught his attention" rather than just what he saw.