When you're worried about be perceived as fragile or emotional...
"I'm getting promoted - yay! The woman formerly in my position was famous for being needy and having emotional breakdowns, and is leaving to find a less stressful job. Over time our boss had given her several allowances to make her job easier given how stressed she was. I want to bring a new sense of calm and levelheadedness to the position, so I'm worried if I ask for all the allowances she was given, I'll seem just as fragile as her. But I'd love the perks! What should I do?"
We all want to be liked.
Congratulations on the promotion! Promotions mean more money and more responsibility but they can also mean more stress.
Your predecessor chose to leave for a less stressful job even with these "perks," so depending on how burdensome some of the concessions your employer made for her were, you should feel comfortable asking for the same benefits. It is understandable that you want to set a new precedent, but you also want to be sure you are setting yourself up for success in what sounds like a stressful job. If asking for these benefits will help you be calm and levelheaded, then you are implementing the self-care needed in order to be successful. These are not "perks" and you will not be perceived as "fragile" if you are taking care of yourself so that you can do well. The good news is the company is already used to whatever concessions were made for her. You're not asking for more than what was already given.
The fact that your boss promoted you means he or she believes you are capable of doing the job well. Don’t worry too much about the baggage of the woman that came before you. Making these requests means you are approaching this new position with your personal wellbeing in mind, and setting yourself up to tackle the challenges of this new job with the level of calm and levelheadedness you are hoping to achieve. Best of luck!