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 ...
Objectively: no idea.
Waste zero seconds. Make sure I have boots on. Run to the edge of the fire.
Simultaneously shout some orders at some of the campers (if they allowed this, they are obviously incompetent, probably also frozen in panic).
Furiously use mu hands to rip fuel (dry branches, pine branches, high grass, dry brush) and throwing it ...
even though my bag is rated well below the outside temperature
I soon become dowsed in sweat
I concur with the accepted answer that a cotton liner would be nice.
Most sleeping bags feature double zipper operation. Do unzip the bottom part a bit. This not only aerates your feet but also provides a way for air to circulate through the bag.