None of your options is really bad (although I would not use the present subjunctive, and probably wouldn’t have occasion to use the present simple in this context either), but none of them strikes me as particularly natural either. If I’ve understood your meaning right, I’d say “She is dreaming of him writing her.” And yes, in American English, the “to” is optional. It’s not a case of leaving it out — in Old English, the case system made the word “to” unnecessary.

Like Dave, I would always say "to her", though maybe in North American usage it is OK to leave out "to". 1. She is dreaming that he wrote to her. This is a correct sentence and means that in her dream, he has already written to her. 2. She is dreaming that he would write to her. This is a correct sentence and could mean that in her dream he used to write to her. ("Would" can be a synonym for "used to" if referring to past habitual actions). I wonder if you intend to say that she hopes that he would write to her? This could be expressed with sentence 2, but it sounds clumsy to me. 3. She is dreaming that he write to her. This is not something I would ever say. I suspect it might be a correct usage of the subjunctive, but it's not usual in my experience.

She is dreaming that he would write to her.

She is dreaming that he would write her