For reference on the no training side:

Dr. Dre and Dr. Venture: Both possess honorary doctorates. The former from UCLA, the latter from a Tijuana community college.
Dr. Pepper: Charles T. Pepper did have a medical doctorate. He is cited as a possible source of the name.
Dr. Evil: Evil medical school.
Zoidberg: Claims he lost it in a volcano, more likely a art history degree.
Dr. Horrible: Likely physics though he claims Horribleness (though it could be a catchphrase.)
Dr. Seuss: Intended to get a Doctorate of Philosophy, got an honorary doctorate of Humane Letters.
The Doctor: Per This - While pretending to be the Doctor in an effort to save herself from an army of Cybermen, one of the planet’s deadliest enemies, in Death in Heaven, Clara said: ‘I don’t even really have a doctorate. Well Glasgow University, but then I accidentally graduated in the wrong century.’

I feel like a fair amount of those on the trained side aren’t medical doctors (Phil, Ock, Robotnik, Lector, Honeydew) or at the very least, aren’t folks you’d want around for your heart attack.