I don't know if you are a PTO or a PTA- but your by-laws should give you the answers to some of your questions.
Most by-laws state the minimum number of board meetings you are required to have per year nad the minimum number of general membership meetings per year, and what establishes a quorum (at both meetings).
Board meetings do not need to be open to the general membership(they are BOARD meetings). If you want to open them up- that is your decision.
Your genaeral membership meetings DO need to be advertised (in some way)- depending on your by-laws (again) most are 7-10 days prior.
Board meetings notice is not usually given in the by-laws- but check anyway to make sure you do not have "notice" number of days for that.
Most of the time it is a week. (I have mine on a monthly schedule- that tkes care of that).
Special meetings are defined as "extrodinary meetings not in the regular schedule. " I.e. if an issue in your area calls for a special discussion or emergency vote- you can call it. But it must be advertised (usually for 3 dAYS PRIOR) AND A QUORUM ESTABLISHED AT THE BEGINNING FOR ANY VOTES TO COUNT (THE QUORUM IS DEFINED IN YUR BY-LAWS).
As for your secretary sitiing in on your meetings with the principal- that is something YOU get to decide. These meetings are usually informal- no minutes need to be taken. You will take any notes you need to make. Now if your principal comes to your board meeting and makes a "state of the school: principal's report- that must be in the minutes.
It sounds like you need to have aprivate talk with your sevretary. Conventions are great, and there are reasons for set rules for jobs- but if your school is more informal- she may need to relax a bit. Let her know what your expectations are- do you want her in every committee mtg? (probably not...) do you want her to make your agendsas? etc.
By the same token- a good secretary can keep you "on task" if a meeting is getting off on a tangent.
BY talking with her- you and she will soon be working together and complementing each others' srengths.