"taller than me" is definitely colloquial English, but textbook English grammar would be "taller than I", or "taller than I am."
I believe "me" is the oblique case, which is grammatically correct here
"Taller than I" is correct, and should be accepted. "Taller than me" is also correct, and much more commonly heard, in my experience. I'm not sure why my earlier comment on this was downvoted so much, but here are a few links to discussions of this question: http://www.grammar-monster.com/lessons/than_I_me_than_he_him.htm http://writing-skills.com/is-it-taller-than-i-or-taller-than-me/
Genitive, if you say выше меня. However for all personal pronouns genitive and accusative cases are the same, if i'm not forgetting something.
It's not clear with я/меня, but substitute some other noun, for instance, дерево, and the correct way to say it is выше дерева, so genitive. The accusative case would have been дерево. I know this probably doesn't help when you are learning, but this can help down the road when you already know it's genitive and are trying to check yourself.
If you mean that she's 180 cm tall and you're only 175 cm, then you have to say "taller". Your sentence works if you are standing on the floor and she's standing on a chair.
...but it will be the same word in Russian: выше -- for both higher and taller than I. You can also say выше меня and выше чем я. Almost the same meaning, although I would personally tend to say выше меня for taller and [она сейчас] выше чем я for higher. The context usually sorts it out.
I is the subject of implied verb, it's incorrect to use objective pronoun.