Sounds like you are in a power struggle with your daughter, and, yes, this is typical of a 3-yr-old behavior. Maybe some of the following will help:
The worst time to have the "battle" so to speak, is in the heat of the moment. Research shows that when emotions are high, intelligence goes down, i.e., there will be no rational solutions on either side. Instead, have a conversation with your daughter about all of these things when emotions are NOT running high, i.e., not in the morning when everyone is rushing.
1) First, make sure she is ok with going to pre-school and leaving you. Openly talk to her about it--just not in the morning. I actually had to ask my son if he missed mommy at school. When he said yes, and started crying, I knew he really did. So we made a "kisses" pact--I would give him lots of kisses to put in his "shining heart" and then when he missed mommy during the day he could put his hand on his heart and know that mommy's kisses--and love--are always with him. It helped. Talking with your daughter about school also may reveal other things about her experience there that may need to be addressed. Maybe there is someone in her class that teases her, or won't play with her, or who always takes away the toy she is playing with. These little things can cause stress for a child and make them procrastinate going to school. A bit of inquiry here can go a long way. (BTW..I often use bedtime--when I am laying down with my son, lights out) as a time to say, "Is there anything you want to talk to mommy about before you go to sleep?" I am often amazed at the things he brings up. Yet if I didn't ask, he would keep them to himself.
2) Have a conversation with your daughter about the morning routine at a time other than during the morning routine. Emotions are already running high in the morning....so you need a plan that you and she have already created. If you involve her in creating the "morning plan" she will feel empowered about it and have more buy-in. Right now, it's "your plan," and the only way she can feel empowered about it is to resist it. Instead, on a weekend, make a game out of creating the "morning plan," Have her make a list of all the things that she needs to do to get out the door in the morning, e..g, potty, breakfast, brush teeth, get dressed, put away jammies, comb hair and so on. Then take a picture of her doing each activity. Then mount the pics on a poster board or bulletin board, having her decide the best "order." Apart from brushing teeth, the order really doesn't matter. She can then draw on the board, write her name, etc. Place it in a conspicuous place--her room, the kitchen or bathroom.
Now she "owns" it--the morning routine. She created it, she is in every pic, and she will be feeling empowered over her own destiny, so to speak If she resists in the morning, remind her that this is HER morning plan...and refer back to the poster board/pictures. Part of your discussion can also be about consequences, i.e., you asking her, "what do you think should happen if you don't follow the plan in the morning?" Lose a privilege, for example. You may need to help her with this.
As a side note, there have been many convos about using "stars" or "stickers" on a chart as a sort of reward. Some are totally for it, others say it doesn't work over the long term. Something you can research....
3) Assess your child's personality disposition. Some kids are incentivized through rewards, some by competition or games. Mine is the latter. If I say, for example, "Ok, mommy is going to get dressed now. I don't want you to beat me. No getting dressed until mommy comes back." It is a sure fire way for him to get himself completely dressed and waltzing in the bathroom announcing "I beat you!" And then he laughs hysterically. Works every time--for my child's personality.
4) See if there are other ways for your child to feel empowered about the morning routine, e.g., pick out her clothes the night before, get her shoes and socks ready in a special area, and so on. Again, the idea is to help her (and allow her) to make decisions so she can feel powerful and strong. When we make all the decisions for our kids, they will try to find power other ways, e.g., by acting out, pulling a tantrum when you least can afford the time, etc. I read a great book on parenting that said that ALL of these behaviors are meant to get our attention to let us know that one or more needs of the child is not getting met. Frustrating as this sounds, when my son had a total meltdown in Target, I had to ask myself, "What is going on in his world that needs attention?" I am always on the lookout for ways for him to feel empowered (without sinking the whole ship, of course).
A great blog for more info on these kinds of ideas is:
http://www.bonnieharris.com/
At her site you will find her blog.
I sincerely hope that there is something in here that will make your mornings go more smoothly. Parenting is the hardest job we will ever have!
And btw...I am an older mom, too, so I understand where you're coming from.