What I should do with the previous day's fire logs which are not fully burned yet? I believe they can be used again, but should they be put on the top or the bottom? Can they be use as a fire starter?
Partially burned logs in my experience do not work well for restarting a fire but once you have a fire going, they burn just fine. Depending on the size of the sticks, I might use it as part of the structure in building my new fire. In general, when building the next fire, the partially burned stuff is on the bottom.
It depends on the resources available. Wood may be thick or thin, wet (green) or dry. You start the fire with the most combustible material, i.e. thin/dry, and then make a judgement about what to place on the fire from the available material.
My techniques when camping at a fixed place are:
thin: make sure there is some kindling dried out ready to start the next day's fire, by placing it near the fire but not so close as to burn.
thick: no need to cut long branches, just feed them into the fire from different directions. Then the next day when the fire has caught, resume with what was left of those branches, which will catch more quickly than the first time.