If the coil is firing even before wicking, it's not the coil persay.
Going over my past experience with newly learning rebuilding, it almost sounds like there is no real issue (besides cracked glass)
One side of the wick is brown but the other is not? Could be a sign that your wicking job is not perfection yet, and the wick is uneven(too thick on one side). or that when covering the juice holes in the rda that you apply to much pressure deforming the wick enough that it's not every flowing juice.
You keep mention a burning taste? Is it an actual burnt taste or it tastes like cotton? New wicks need to break in some, depending on brand/material/quality, some have a taste that stays or dissappear after use. I use regular cotton balls from the grocery store, and personaly it takes around 3 mls of juice before the cotton taste starts to dissapear.