I put in "I'm happy with you" in a word selection exercise, and it accepted it as correct. I wonder if the problem for you was the "I am" rather than "I'm"? If so, that should be reported.
I entered "I'm happy for you." Is that incorrect, and if it is, how would you say that in Hebrew?
It's incorrect. I'm happy for you (plural) in Hebrew is: אני שמח בשבילכם, or אני שמח עבורכם.
Is the word order important? I had the listening exercise for this question so I didn't have to write the translation. However, isn't it more correct/common to say "אני שמח איתכם"?
It's so hard for me to pronounce איתכם without accidentally triggering a trill to sound like איתרם, any tips?
Well, unfortunately the suffix כֶם is always pronounced as a fricative, even when it follows a closed syllable. You may pause shortly after the [t] in order to let your tongue leave its dental position, until you learn to say it faster. By the way, do you pronounce the רֵישׁ as a trill too or as an uvular approximant, the throaty r, so that the accidental trill is not part of your usual sound set you use in Hebrew?