Problems with my mom over birthday?

In spite of your problems, your mind is on the right course; you asked how to set boundaries with your parents.

There comes a time in life when it becomes necessary to assert ourselves to former authority figures, because we become our own authority. When it comes to setting boundaries, becoming assertive is the first step, and being careful to not become overly aggressive is the first (and arguably longest) obstacle to overcome.

You are 24 years old, your primary focus as an adult living with your parents is to make your place in the household so that you are earning your keep; so hammer out your household responsibilities and then exceed them. As your parents become aware that you are an asset around the house, they will support and respect you more and resist you less.

When it comes to your mother and her mental illness, mental illness requires compassion and patience, this is true, but not at the cost of your own life. You must be patient with your mother, but when you have something planned and organized, you do not deviate from it simply because your mother has a tantrum, and if that involves her not coming to your birthday party, then so be it, and do not be afraid to say as much to her. She is already blaming you for her feelings, which is a huge red flag on the abuser scale. Maybe if she sees herself being less-involved in your life, she will be more prone to seeing a professional for her problems and beginning to fix them instead of staying a victim.

Asserting oneself means sticking to a plan or a decision in spite of opposition or consequence. Leave your mother at home and go enjoy your birthday party and let your friends pay for themselves. If your mother throws a tantrum, tell her you're not listening to it and go somewhere else. Assertiveness can require a certain amount of detachment and dismissiveness.

Be responsible with your course of action, discuss your doings with your parents so they are up-to-date on how you're handling things, this keeps them involved and displays responsibility on your end. When one of your parents opposes you without justifiable reason for doing so, such as your mother's case, do not argue with her, simply go somewhere else and leave her to her sour mood; you are free to ask her later, when she is lucid and calm, if she would like to talk about it.

If you have any more questions, comments or concerns, my inbox is always open.

Always remember that you deserve the same chance at pursuing your own happiness as anyone else. Never allow anyone or anything to stifle your will to live happily, and most importantly, never forget that you matter, and you are not alone.

Oh man, I didn't realize she was supposed to come to the party until you said it.
Uhm, setting a boundary would be telling her she is welcome to the party as long as she will not bring up again the payment part. If she throws a tantrum there, I think you can calmly tell her how you already discussed it, and to not ruin the party, and changing topic.
She might try to pay the whole thing, I don't know if you can have a word with the cashier about it to prevent it.
Setting a boundary can also be leaving her home if she still refuses to come to your party until you apologize or whatever other drama she can put up on that day. If she wants to come and is ready on time, good, otherwise you leave without her.
I don't know if any of this examples apply, but in theory, you have limit how she will affect your enjoyment of your birthday. Your dad and sister seem on your side, that's good.

If she stays home, should your dad stay too to check on her?

I'm sorry I meant, that on Saturday i'm going with my friends on Sunday we are just having a family dinner at a restaurant of my choice. It is important to me we spend the day together because I am moving to Miami for medical school and probably wont have another birthday with them for another 7 years