Hooboy. I remember the other thread about your co-worker.
It sounds like you both just bring out the worst in each other. She makes you respond like a first grader, and your presence makes her act like she's 5 (hiding/taking your pens? I mean, c'mon...).
As the famous phrase goes: You can't change her, but you can change the way you respond to her.
The best way to deal with her is to be as professional as possible. Do not respond to her without first filtering your words to yourself. I had a colleague like this, and I always typed out what I wanted to say in Notepad and edited it before talking to him. If it was something that required an immediate response (like to a direct question), I always "postponed" answering it by saying "Let me get back to you with that after I'm done with <insert whatever it is that I am doing>.
I like your approach with the pens. If you see her with any of them, just politely say "Janie, that looks like my pen. May I have it back please?". If she makes a snarky comment, just smile and say "Okay, but I would still really like my pen back".
Same thing with the apparatus. Don't get angry (or even if you do, don't show it). Just say "Janie, I have already booked that apparatus for <insert the period of time you need>. You are welcomed to use it after I am done." If she says things like "You are doing something else!" Just repeat with "Janie, I have booked the use of that apparatus and I am afraid that you are going to have to find other alternatives if you wish to use it now". If she insists on sharing, just tell her "No thank you".
My colleague used to be very childish - he would have probably refused to budge and just ignored me. I have so many stories about him. For example, we had a parallel computing setup which I was using (for a simulation that would take a week to run) and he pressed the cancel button on Day 5 so that he could free up the PC for his use. Or another time, he called the purchasing department to cancel my equipment order and sent an email stating that my fund was actually his, and then proceeded to place another order, using my funds! I found out when the finance manager called me to check on why the data is inconsistent! Then when confronted, he insisted he never did it, and that the purchasing department was setting him up!
Oh Goodness, the stories I could tell, but oh wait, I'm derailing the thread
Anyway, my point is that Janie doesn't sound as awful as that (I hope!) She just seems to enjoy pushing your buttons. So stay polite and don't respond and take pleasure in the fact that the more professional you behave, the more childish she may get to try to push your buttons. And the more this happens, the more the situation will turn to your favour. Currently because you are both sniping at each other, your supervisor will see it as "two people quarrelling". When you are as professional and polite as she is childish and annoying, believe me when I say that everyone will call a spade, a spade.