I'm going to take some times to answer all these question since there isn't a single answer to fit each situation.
For MM days, it's 50 minutes before the normal park opening. So today when the park opend at 8am you could get a 7:10am reservation. Works well because you have 50 minutes to eat and finish and get into the park without a MM credit.
For non MM days it varies. For early open, or long hour days it opens when the park opens. So IMO a waste to be eating when lines are shortest. For short hours days like last Wednesday or next Monday it can open 10-20 minutes early, some have eaten 30 minutes early but I'm not sure they do that anymore since I haven't been able to get a MM day reservation that early in the last year. So again a waste because you won't be done in time to get into the park.
I called the DL Dining phone number last week and asked what time the Minnie Character breakfast takes place, and she said from the time the park opens until 11:30am. I didn't ask for more specifics because it's still over 60 days before we will be there, so I'll call back within the 60 day window when I can make our reservations. If what she told me is true, it sounds like they only serve breakfast for one-and-a-half hours on days when the park opens at 10am.
We will be at DL for 4 days in Nov (Tues thru Fri)...3 of those days the park opens at 10am and the last day they open at 8am. I'm debating whether to make a reservation on one of the 10am days, or making a reservation for 10am on the day when the park opens at 8am (that way we can enjoy two hours in the park when the lines are shorter, and then have breakfast...but then I worry because I've heard the characters start leaving before the 11:30 end time).
But, given the information Skiingfast just provided, it sounds like we could make a 9:10am reservation on the MM Thursday (we would use our MM on Tuesday), and we'd have 50 minutes of eating/character time before the park even opened. How long should we expect a character breakfast to last for our 4 year-old daughter to get some good character interaction?
I just worry about using any early morning park hours eating breakfast. There already seems like there is too much stuff we need to get done in the first hour or two of the four mornings we will be there (Star Tours, TSMM, The Little Mermaid, getting WOC fastpasses, Soarin', and any other rides that are much easier if done early in the day).