Though the roo mi in sanskrit does mean to fix or establish, the word mita in that sense is rarely recognized today: the masculine form was indeed used as a name in the ancient times, but I do not think the modern feminine mitA derives from that. Certainly, the name is usually interpreted as a
NIA word which originates in the sanskrit mitra, a friend, probably connected with mith, to unite.
I do not know about Rosmita, but would guess it is a made up form patterned after names ending in smitA, smiling, from the root smi cognate with and meaning smile, or to blush; it may also be influenced by rashmi, an old word originally meaning rope, but the persistent metaphorical use of ropes of light have pulled a lot of rope words into ray words in Sanskrit, and the ray of light is the only surviving meaning of rashmi. Of course, it could also be a semitic word borrowed in since the sound structure is very familiar.