I am building an icon that has a brick wall. So I set the grid lines at certain distances so I could draw the lines between the bricks.
Seemed simple enough to then just connect with the line segment tool. But when I zoomed in, the line was not being drawn exactly where I wanted it to be. It was not only away from center, but keeps getting skewed a bit.
Ive struggled with this as well. What I found to be very helpful is setting the grid in illustrator to be a 1px grid. Then you can enable snapping to grid. This helps but sometimes I find its better to bring Photoshop icon mocks into illustrator after I have made them and saving from there.
Also I see one thing that is throwing you off. One reason things are not aligning for you is you have stroke enabled in your pic. You want to turn that off and only use fill.
This article helped me out a lot: