Setting limits for a one-year-old drama queen?

Every time I try to stop my one-year-old daughter from doing something, her bottom lip starts quivering, tears start flowing, and she looks at me like her heart is broken.

The things I'm trying to prevent aren't frivilous--don't come near the oven when it's open, don't pull stuff out of the trash and eat it, don't crawl in the cat litter bin. You know, stuff that she really CAN'T do.

Plus, I'm not being rough on her. I just say "No ma'am." and pull her back or try to redirect her.

But she's acts like I've killed her favorite kitten or something. I don't know how to respond. Do I hug her and reassure her that I do love her? If I do, is this sending mixed messages that I didn't really mean she couldn't eat the cat litter? Is all of this drama for real or is she just testing boundaries?

Her older brother never did this with me, and I'm just completely confused about how to deal with this one.

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You aren't going to ruin her life if you stick her firmly in the play pen or put up a gate between kitchen and living room. She knows what buttons she is pushing, You aren't being mean by keeping her safe. That is your job, Mom. She will not publish her memoirs and tell the world about her mistreatment at a year old. She is a baby. You are the adult. Deal with it.

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She just might be over sensitive. I'd wait a minute and then go over and hug her and explain lightly why you removed her. The oven is hot, ouch! Mommy doesn't want you to get owies.
When my daughter gets upset over something like that, I just calmly say something like, "I know you want to touch that but you can't because it's dangerous. You can cry all you want but it's not going to change anything. You can not touch ---" If she's still crying, I sit her in the next room (in my sight) and tell her I hope she gets over it soon so she can come play with mama again. Then I ignore her until she gets over herself. Of course, I'm still keeping an eye on her, but I'm not feeding the monster, so to speak. Without an audience, she gets over herself pretty quickly. They're all different, of course, so that might not work for you. I know it's tough as they start to test those boundaries. Good Luck.
hmm this is a hard one. She may be very keen for a one year old and is already testing you (early) or she may just be genuinely upset that she's done something wrong.

I seem to remember my son getting upset everytime I said no to him, but something has to be done about those types of things that they really just can't do (like you mentioned). I myself would tell him no and then sit down and talk to him calmly even though I know he didn't understand alot of what I was saying I just wanted to get my calm voice across to him so he'd stop crying without rewarding him with hugs and kisses (exactly as you said). I'd hold one of his hands and point at the oven (or whatever it was) and calmly talk about it until he was also calm, then redirect him towards another activity.

I think for that first bit you just kinda have to keep at it, she'll learn no doesn't mean she needs to get upset only that she's done something she shouldn't and that will take time for her to understand. When she hits her terrible two's you know she's at the point that she can understand that it's bad and that's a different story. 1 however, is just too young in my opinion.

Good luck and keep at it, also don't let anyone make you feel bad because she starts crying when you tell her no as I noticed one older woman try to do this to me in the grocery store once and she was completely in the wrong. Besides, it's easier for other people that are not her parents to try to buy her off with candy or acting like nothing happened, and that is not the idea you want to reinforce.
Set the limits and don't back down. You can tell her what's wrong and then tell her you love her just don't let her turn around and do what you told her was wrong. My daughter is very sensitive you tell her no and she will whimper and cry. I let her be upset for a minute then I pick her up and tell her she can't do whatever because and tell her why. Then I distract with something else and we go back to playing. She get's the point and knows she is loved.
Seems like you already have gotten a lot of good advice here. From what you have written about your little one it appears she is quite sensitive especially to correction. But this doesn't mean you should compromise.

You have to stick to what you know is right and safe and then give her the reassurance that you love her very much and care for her. Love never fails....!
My son did the same thing until I started to tell him (after I told him not to do whatever) "I'm sorry that you are disappointed but Mommy doesn't want you to hurt yourself. So, you can cry if you like, but it is not going to make me say yes." Once he got a bit older, I would add... "you can stomp, you can yell, you can throw yourself on the ground and cry like a baby... but it is just not going to change anything". He would laugh and then be fine.
Instead of telling her what not to do, try redirecting her toward a different, more favorable activity. At that age babies respond better to positive direction. For example, if she's heading toward the litter box, tell her, "Let's play here with this..." and hold up a favorite toy.

Her behavior is very normal.
She probably doesn't understand WHY she can't do these things. Think about it. She's learning about the world, about herself, about you, and she wants answers. Continue to do what it is that you are doing, but also give her a reason WHY. When she tries to go near the stove, tell her "No, that's hot". Show her what hot means. (I'm not suggesting that you allow her to burn herself! Show her hot and cold with running water. Put her hand under cold water, and then put her hand under very warm water... ) She's still a baby, and keep in mind that she is still new to this world. She's just now learning how to deal with her own emotions and yours as well!