1. Should I consider myself single or separated? We were never married, but have lived together for most of our relationship. FWIW - I consider separated to relate to marriage, but after reading some posts and talking with some people, I am not so sure
You are neither single nor separated. Until you are single, living outside of the same household ~ you are in the vast middle-of-the-road.
2. At what point, if ever, is it acceptable to start meeting people and dating while continuing to live together with the ex? Is it disrespectful?
That would be between you and those that are willing to meet/date someone in your situation.
3. How would you react if you met someone in a similar situation? Would you consider dating them?
I know someone here on POF in that situation. It wouldn't work for me personally, not because of the living situation but because they both seemed in such a hurry to replace one another. I think anyone leaving a long-term-relationship should spend a year or so re-inventing/evaluating their life, the loss, the role they played in the failed relationship, etc. JMO