"Ella ha llevado el mismo sombrero durante un mes."
Translation:She has worn the same hat for a month.
Nope, no reflexive required for "llevar". So if you're trying to say I wear/I'm wearing my coat, then it's fine. Saying "put on", however, requires the reflexive "ponerse": "Me pongo el abrigo"=I put on my coat. Btw I don't think that the possessive "mi" is needed because it's assumed that the coat is yours in this case.
You just have to learn them along the way! Sadly, there's no shortcut for knowing if a verb is reflexive or not. http://spanish.about.com/od/sentencestructure/a/reflexive_verbs.htm
I completely understand, DOPs and IOPs take time to get used to, I still have trouble with them every once in a while. You just have to practice and get used to using them! These links might help: http://studyspanish.com/lessons/dopro1.htm http://studyspanish.com/lessons/iopro1.htm http://studyspanish.com/lessons/iodopro.htm
You employed the present perfect continuous tense, rather than the present perfect ("has torn") tense. I also don't know if it is equivalent in Spanish or not, but is certainly a subtly different meaning in English.
A Google search on "English verb tenses" gives interesting hits.