If I have an academic dismissal from a school should I ever go back there?
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.