"I have a small nose."
It is just another style to state it. Everybody has only one nose normally.
Yes!! When mentioning family members & pet animals, we can skip 的 . Example can write 我爸爸 instead of 我的爸爸。 Any suggestions by native speakers please.
nope.when we need to stress "我"，we add"的"，if not we usually drop it.and "我的"sounds more official