I'll give it a shot, because I consider myself a relatively empathetic person:
> 1. How did you develop your listening skill?
In my mind, there are at least two different types of listening that I can think of. Let's call them "surveying" and "socialising":
a) When you're "surveying", you're taking in information to create some kind of output in the end. This could be listening to a co-worker describe a problem, or it could be listening to a friend asking for relationship advice. What's normally the most important here is to indicate that you are in fact listening, and resist the urge (if you have it) to break in during the explanation, unless you need clarification. At the end, try to summarize what you've understood, not just to indicate that you actually listened but also to clear up any misunderstandings.
b) When you're socializing, what can lead to conflict is if you approach it like you're doing "a": this took me years to realize in romantic relationships, that a lot of complaining is not someone asking for a solution. When meeting people in general that's pretty obvious, people like whining about things just to agree on things that suck so you can bond over it, but for some reason I always assumed that in more intimate environments you'd probably bring it up to get some advice. Well, that's not true. So if the person is just airing grievances, saying "that sucks" and doing some whining on your own that's tangentially related might seem like a really bad solution, and it is, but noone is asking for a solution, so don't offer one.
> 2. How to be more empathetic?
Having empathy is quite simply the capacity to relate to another persons feelings. That doesn't just mean "how would I feel", it also means "how does this other person differ from me, and how would I feel in their situation", which is something completely different. Other people will frequently feel insecure, just like I'm sure you do, and it's not always we consider that.
It's also important to point out that it's impossible to care about everybody's problems, but the important part is not genuinely caring, but dealing with the situation in a way that is socially acceptable. You know, like walking away while someone is talking about something that's really weighing on them might make sense from the perspective that you don't honestly care, it doesn't make sense from the perspective that you don't want this person to feel rejected and meaningless.
Like all skills, empathy takes practice if you're not used to it. Maybe you're from a home where talking about feelings wasn't encouraged, regardless of why you feel you need to be "more" empathetic, don't be too hard on yourself. Honesty goes a long way, and if you feel like you don't know how to act in a situation, it's a million times better to be honest and open about how you feel than to fake it.
Most of the time, showing empathy is just acknowledging that you're seeing another person and what they're going through. 15 minutes of heartwrenching details about a breakup doesn't need a 15 minute response, it's well enough to just acknowledge that the whole situation really sucks and you're there if they want to talk about it some more. A lot of problems don't have solutions, and empathy is meeting someone in that uncertainty.
> 1. What common failure modes do you hit in your relationships as a low empathy person?
I used to be a pretty low empathy person, but I think honest and open communication goes a long way to rectify that in the long term. As long as you're open about your weaknesses, people at least know what to expect and what to tell you. If you tell someone that you need them to be extra clear on what they mean or expect, and that your "low empathy" behavior isn't meant to cause offense, at least people close to you will probably feel more comfortable in correcting it. If you don't, most likely nobody will say anything, because adults normally aren't in the habit of raising eachother.
> 2. How do you avoid them?
If you want to avoid a behavior that you consider unwanted, that's going to be hard if you yourself can't identify them when they happen. So I would say that a good first step is probably acknowledging the problem, and trying to identify it when it happens. When it does, despite what you have in your mind, try acting in another way and see if it gets you a better result. If you can't come up with anything and you need help, ask for help. There is no shame in being open about what you don't know and what you're not good at, that's the first step to self improvement.
One quote on empathy that has stuck with me is that we "always judge others by their actions, but ourselves by our intentions". We tend to be less forgiving to others, and one way out of that may be to try to find the underlying intention rather than what that person did. Empathy is, after all, the ability to envision yourself in another persons position. In the long run, I think this leads to a lot less conflict as well.