Note: I'm not sure what version you're playing - if you have Gods and Kings, or Brave New World - both expansions introduced some differences to diplomacy, so what I say next may not apply. I have everything up to BNW, so...
If you look at some of the things that can turn other civs against you the list reads a lot of things that are just the result of you playing the game. "They covet your territory", "they covet wonders you have built", "you are spreading a different religion", "you have chosen a different ideology" "you are competing for favor of a city state", etc.
Yeah, I get those in the early game too. The key is to try and make friends with at least one civ. Then they're more willing to overlook some of the minor negative points like the wonder thing or the city state influence. In my current game as Venice, I had neutral status with everyone. Some positive points, some negative points. So I made a few trades of spare resources. But the big turning point was when I liberated an Indonesian settler from the barbarians (+ points). A few turns later, they came to me with a Declaration of Friendship. Which gave me + points with the Netherlands since they were friends. Sure enough, several turns later, the Netherlands came to me with a DoF too. After that happened, the city-state influence and wonders negative points from both of them went away.
Long story short, find a way to make a friend.
(On the flip side, both China and Germany are now friendly to me too, and I have no idea why since they're not friends with anyone else and I haven't done much with them.)
If I stay well armed for fear of an attack they ask if I'm planning to attack them, if I don't move my units away from their borders I'm called a liar,
Yeah... don't keep units on their borders. If they call you out on it and you say that you'll move them away, that pretty much locks you out from war for a long time otherwise you'll get negative points for breaking a promise.
if I defeat every enemy that enters my territory we often remain at war indefinitely until I retaliate and capture or destroy some of their cities and stongarm them into a treaty, which assures continued war with them or the other AI civs in future rounds.
Defeat their units but try to avoid taking their cities. In BNW, the warmongering changed so that it is based on taking cities. More than that, it's relative to the size of the target. So if you take 1 city out of 10, it won't be as bad as taking 1 city out of 1. And you can reduce or eliminate the warmonger penalty by liberating cities that your target civ took from others (city-states, other civs, etc).
At least, that's how I understand it from reading several threads on the subject on CivFanatics.