Hey Rich,
Sounds like you're setting up a lot of confusion for yourself here by trying to place yourself on a map which you're not familiar with. It's no big deal, everyone does it but the best way to really understand it is to do it for yourself. I'll try to answer a few of your questions here and see if it's helpful to you.
I have read descriptions of people becomming over-whelmed (sic) with pain and then the next moment "dropping into equanimity." But what does that mean exactly and how do you do it? What actually happens?
The pain referred to here is most likely mental anguish or emotional pain (common during Dark Night but certainly not always the case) being experienced in "Re-Observation", or 10th ñana, which is the insight stage immediately before Equanimity. To move from 1st to 11th ñana and on to Fruition you just need to apply vipassana/noting, if you do the practice correctly and with commitment then you will encounter these stages for yourself. Dark Night sucks, admittedly, but it's just a stage along the way and a yogi needs to learn how to navigate their was through it which comes with practice, and moving into Equanimity is a pleasure after the harsh, jagged vibrations of 10th ñana.
I am familiar with the experience of a negative emotion arising... such as rage. Often it'll come with an image, maybe some sound too. Normally it's strongest when it first arises and then gradually goes away, but it can get stronger if accompanying thoughts and images arise later.
Good! Apply that to the positive emotions too, but this time try to observe the entire sensation from the moment you notice it to the moment it vanishes. Watch how the bare sensation arises first, then the feeling tone, then the mental aspect. Pay attention to your sensate experience, this is where the answers are!
However I have also noticed that the same feeling can later immediately arise again even as strong or stronger than before.
Even better! This is knowledge of the Three Characteristics, 3nd ñana, keep doing what you're doing and get a practice report up and running, that way it'll be easier to offer more specific advice.
Does this even have anything to do with equanmity?
No.
3rd insight knowledge, Knowledge of the Three Characteristics. You're seeing impermanence in real-time, pay attention to this and see how every single sensation is subject to this simple fact.
Is it a constant effort to be in equanimity?
No. Forget equanimity for the moment, you'll come across it sooner or later if you practice well.
Is being in equanimity related at all to your concentration, speed of noting, ability to focus etc? Or is it unrelated to these "skillful" aspects of meditation completely?
I don't want to confuse you further by going into samatha and vipassana jhanas, and other more technical stuff. To keep it simple, getting to the 11th insight knowledge, equanimity, takes practice and solid technique. It's nothing special, it's just another insight stage which you will encounter is you practice well. It's nothing to do with the speed of noting, noting takes you through the stages of insight and requires concentration. It's worth remembering that insight can produce concentration, and concentration can produce insight, they're not mutually exclusive.
I understand I'm asking a lot of questions here but if anyone has some answers for me I'd appreciate it a lot because this is something twhich has disturbed me probably for years now.
Well, time to stop wondering and tell us what you're doing during your practice right now! Your posting after this original one is MUCH more like what you're looking to post about, this is real-time phenomenological stuff, not just speculation so you're obviously putting the work in here. Nice one!
Look at what it is about this which is causing feelings of disturbance to arise. Ask as many questions as you can, it's always good to get more than one opinion but a solid practice schedule with good technique will answer most of what you're wondering.
I'll reply to your practice notes directly rather than cluttering up this response.