Common phrases (in Am. English)
"She fell on her head" - she fell landing on her head before any other parts of her body. "Landed on her head" is used similarly, but was used less frequently until ~2006.
"She fell head-first" - she fell with her head oriented downward/toward the object
- This does not necessarily imply that they landed head-first. (e.g. They fell head-first towards the water, then flipped and landed in a pencil dive.)
"She fell and hit her head" - she fell and her head hits something
- This does not necessarily imply the orientation of her fall, or that her head hit first. (e.g. Perhaps she landed on her back, and her head hit next).
- This does not not imply that her head hit the floor. (e.g. Perhaps she fell on the floor, but hit her head on the counter as she fell.)
"She fell down [a well]" - When used with an object like "well" or "hole", "fell down" implies falling into that object.
"She fell down." - When used without an object, "fell down" generally implies the brief, discrete action of falling from a standing position on the ground. E.g.
- YES: "she was walking, tripped, and fell down."
- YES: "She fell down five times."
Note: the phrase "fell down" is generally not preferred when expressing distance or duration. E.g.
- YES: "She fell for 60 seconds before opening her parachute."
- NO: "She fell down for 60 seconds before opening her parachute."
"She dove to the floor" - She intentionally dove parallel to the ground, landing on the floor. For a visual, picture a sports star diving for a ball, or an action hero under fire diving to safety behind a large object. Unlike "dove into the floor", below, this has been used with some frequency in recent decades.
Other phrases from the question
NO: "She fell with her head onto the floor" - this is an awkward construction, and not one I have personally ever seen used.
NO: "She fell & dived into the floor" - also an awkward construction and not idiomatic. One verb will suffice, and "into" is probably not the right choice.
PROBABLY NOT: "She dove into the floor" - This could be used literally in a (fantasy?) context, where she literally passed into the floor. With care, it could perhaps be used metaphorically, maybe emphasizing the severity of her fall. Both of these are edge cases, however; this is less awkward than the prior examples, but still not a common/normal phrase.