The questions are for your own pondering before I weigh in, but in any case, I feel you.
About people caring - you're right, they don't really care about you. That's a great thing. It means you have a blank slate to create a positive relationship on. It's really all about proximity though - no, strangers don't care, but it's not that they don't want to care. It's just that they don't know
you - they have no real concept of "you" - and it's essentially impossible to care about a non-entity. In some vague philosophical way, most people do care about "others", but once people get to know you individually, that's when they care. Just think about seeing someone spilling their bags in the store - as a stranger, you probably feel bad for them in a vague way, but may or may not help because you don't know them. But if you'd just had a conversation with them, of course you would help them. If you were their friend, you would go even more out of your way. It's not because they're not worth caring for when you're a stranger, but there are lots of other things in your life you care about, too, and if you cared deeply about everyone then you wouldn't have very much time left for your loved ones. It's simply too much for any one person to handle.
It's so important to remember that 99% of people who don't know you are essentially neutral in opinion. They don't know
you, after all - how could they have an opinion on you? But we have
to form judgments about everything to be able to navigate our world, to be able to prioritize and respond, so people form opinions about others based on whatever limited information they have - so people are really going to judge you regardless of whether you choose to engage or not. However, people's opinions of you really aren't as dependent on your own behavior or person as it seems - they're really more about who those people are themselves, and their personal perspectives, since they're filling in all that unknown with personal assumptions.
The other thing is - what if everyone feels like you? I personally have a problem with feeling like I'm infringing on people when I engage in interaction with them - my boyfriend called me out on that recently when I was explaining why I hate calling people on the phone. He said, think about how you feel when someone strikes up a conversation with you, or asks you to hang out. Are you upset? No! Maybe you don't really want to talk right now, or you don't want to hang out, but as long as the other person isn't pushy about forcing you to do something you're not interested in doing, attention is generally flattering. Most people are like you and I, not hyper-competent hyper-successful judgmental people. 4s and 6s both tend to have a native sense of inferiority that's completely illusory, and it makes outsiders seem like they're intimidating. It's so important to keep in mind that is an illusion. They're just struggling individuals trying to make their way in this big world, just like you and me. Now that I'm in my 20s, I have been stunned to see how those "super popular" girls in middle and high school I was always intimidated by have turned out to be completely normal - quirky, even - people that I am really not afraid to talk to anymore, and they even want to talk to me!
I hate rejection too, and I struggle with that, but I've recently really been hit with the hard lesson that people are going to like and dislike you regardless of who you and and what you do... and you just kind of have to accept that there's a possibility - a probability - that some people won't always like you, and there's effectively nothing you can do about that... so there's essentially no use in worrying about them or their opinions. It sucks, but it really says more about them and their personality than it says anything about you. And sometimes there are just bad personality match-ups. I can think of three people off the top of my head that I like well enough in a people-sense, but we do not
get along. At all. I wish them a very happy life, far away from me. Regardless, in every case, the best thing you can do is be the best person you can be, and to do whatever makes you happiest - if that's being more open to people, then engage them more, or be more open to engagement.
As for starting out engaging people you don't really know, just a friendly smile and a hello is a good way to start. Then hello and how are you. Sometimes people will give you a more in-depth response to that question - "I'm so tired today!" as opposed to "Fine!". That's a good cue that you can open more conversation - asking questions is an easy way to start, since most people like to talk about themselves - it is, after all, our single easiest point of reference. If they're not responsive, you can always say "it was good talking to you, but I need to head on to ______! Bye!" If they are, then aim for a mutually-interesting subject. Things should flow organically from there.
You have nothing to be afraid of, really. Take the plunge a few times with people you don't care about and will never see again - someone waiting with you in line in a big store on the opposite side of town, for example. The more you do it, the easier it gets.