If I have an academic dismissal from a school should I ever go back there? As a 19 year old I had to withdraw from a university because of depression. However, at the time, one professor refused to let me withdraw and gave me an F. On the strength of that grade I was dismissed from the university.

I got myself together and went to DePaul and graduated Cum Laude. Now I have received an offer for an assistantship (full ride) and a stipend from the university I had to leave as a young student.

I want to teach at university level and will someday be sending transcripts on to a PhD program from my MA. Will the dismissal show on my transcript? Will it matter? I appreciate you help and input.
Undergrad and graduate transcripts are normally separate things. If you want to make sure whether the undergrad grade would show up on your graduate transcript, that's a question you should ask the school's admissions and records department.

If the F is going to show up on your apps, you could choose to explain it in your statement of purpose, or you could choose not to explain it. A single F in a single course is probably not going to cause anyone any big concerns if all your later work from other schools looks good. If you choose to explain it, you run the risk that there will be people on the admissions committee who have medieval attitudes about mental illness.