To start, God doesn't exist, at least not in any sense expressed by Judaeo-Christian theology.
How do we commit sins? ---- You can't commit a sin. You can only commit an act. Whether or not such act is a sin depends solely on one's definition of sin, which varies widely from person to person.
Is there sin without God? --- It depends on your definition of "sin". I don't think it's possible to "sin" because I don't believe in sin, or God. But I also recognize that certain cultures have developed a set of fantasy rules to live by (aka religion) and those rules state that certain acts are sinful. While I don't personally believe in sin, I can observe ones behavior and conclude it is sinful according the rules of a given religion.
What is it that is directing our conscience? --- Who says our conscience is being directed by anyone or anything? We all have a brain that has a somewhat variable level cognitive ability and a lifetime of extremely variable observations and experiences. Those factors combine to produce a personal conscience that is unique to each of us.
Is it just chemicals reacting in our brain? --- Basically, yes.
If that is the case, why do a bunch of people with chemical reactions in their brains get to decide what is right and wrong for me to do? --- That's life. Unfortunately, there's not much you can do about it other than:
1. change other people's definition of right and wrong (somewhat difficult); or
2. do whatever you do want to do and deal with the consequences (usually minor) that result from other people's chemical reactions.
Bernard