"swoop' means to move rapidly downward through the air.
Despite the downward direction implied by the word, we still tend to couple it with 'down' as in:
"The barn owl can swoop down on a mouse in total darkness."
"The aircraft swooped in to land."
and in the sense, "to carry out a sudden attack, especially in order to make a capture or arrest" : "Police swooped on the house where the drug dealer lived."
It can also have an object, as when used to mean "seize with a sweeping motion" : "The mother swooped up her child in her arms" implying that she bent down quickly and picked the child up.
We talk of planes swooping over an airfield, which really means they dived down towards the airfield, and then rose back up into the sky.
I suspect though that often people will use 'swoop over', as in "On the Queen's birthday, the RAF planes swooped over Buckingham Palace" when indeed they did not dive-bomb the building; but people are referring to the fast movement of the planes without any intention of suggesting the downward motion. So, both 'swoop up" and 'swoop over' should be regarded as informal.
Yes. Look at this sentence:
"The eagle swooped over the field."
The image that this conveys is that the eagle dived from some height, flew at a low level over the field (looking for prey) and then would have flown back up into the air again.