"Genki desu ka?" Is Rude To Ask? What?

I learned something very difficult for me to adjust to today, about the Japanese culture.

It began with a classroom discussion about how Japanese people don't usually ask people "Why?"
"Why do you like this?" "Why do you feel this way?" "Why do you do this?"

It's because they see it as an invasion of their privacy, or maybe as if they are being asked to reveal more of them self then what they are comfortable with.

Then I remembered learning that Japanese people never ask each other how they are doing. Which is strange because English teachers actually do a whole lesson on "Hello. How are you?"
This was actually something that I came to realize on my own, after trying to be friendly and asking my fellow Japanese co-workers how they are doing in Japanese, which is "Genki desu ka?" (Are you well?) Whenever I asked someone this in Japanese, they would laughed and seem a little uncomfortable, and reply with an automatic "Hai, ganki desu." (Yes, I'm fine.)

Then I connected the dots and realized that not only is it not normal for Japanese people to ask each other how they are doing, but they actually view that as rude as well! I was mortified when I realized this because for one thing, I would try to be friendly toward the principle of my school by every once in a while, going up to his office to say a casual, "Hello. How are you?" (I've tried this in both languages.) And he would always answer me with the appropriate words but never look at me when he answers.

The other very troubling predicament I realized was if I couldn't ask people "How are you?," then what do I say to try to be friendly? What Japanese people normally do is, talk about what the other person is doing, or about the situation at hand, to start conversations. BUT I DON"T SPEAK JAPANESE!!! I became pretty frustrated when I realized this. How was I supposed to be friendly if I couldn't ask people how they are doing (The only way I knew how to be friendly.) and couldn't just start a conversation about anything?

And this is where the story ends. I have no solution. Well I have one solution. Learn to speak Japanese, TJ! But until then, I've got nothing. So... what I think I will continue doing is asking people how they are doing in English, to show that I'm trying to be friendly because it's actually more acceptable when I asking the question in English than in Japanese. And I'll try to just start random conversations, even if I the person I'm talking with can't understand a word I'm saying, and I can't understand a word they are saying to me... Sounds like an awful idea... What do you think?