What's the difference between: "اِمْرَأة" and "امْرَأة"?
It's the same word, the only difference is that in the first word the first letter, 'alif' (ا), is written with a diacritic, while in the latter it isn't.
Yes, and the vowel is only pronounced (and marked) at the beginning or after a vowelless consonant. Otherwise, the letter alif is just contracted with the preceding vowel, e.g., وَامْرَأَةٌ wa-mraa2a-tun ('and a woman').
Because Rosa ends with a vowel, the i is not pronounced and not written