It depends. You can have a Homunculus that subverts their preconceived personality type, but make sure they at least have one or two traits derived from their perspective element. That being said, a Homunculus' personality is not set in stone, nor does this lore actively dictate what your character should be. Its purpose is to broadly guide their creation.
Now, for the strengths question: somewhat contrary to what I just said, their strengths and weaknesses are somewhat static. Air Homunculi have strengths associated with the air element, that being of intellect (quick-wits), speed, etc. Fire Homunculi have physical power. Earth Homunculi are much sturdier than the rest, and have a good deal of raw strength, etc.
I suppose you can be a little flexible with their physiology, as long as you don't do it to the extent that it will draw complaints. When in doubt, stick to the guidelines for this aspect.