The mixture becomes less viscous and more uniform in general, but always filter through cheesecloth whenever the solution goes from the container where you keep it to the tray and back. Keep the solution clean and refrigerated when not in use, and it will last a very long time, particularly if you use a little acid as a preservative.
The albumen mixture should be at room temperature before coating.
I recommend floating the paper, both for the albumenizing and sensitizing steps. This takes some practice, and there are a few different ways of doing it, but it was the standard historical method, and people who made thousands of albumen prints evidently found they got the best results this way, even though they must have been aware of other methods used for other print processes. Any kind of brushing of the silver nitrate solution will leave brush marks, which you may want and find interesting as an aesthetic choice, but if you want a uniform coating, I wouldn't do it that way. Some people have been able to achieve a uniform coating with a coating rod, laying down a line of silver nitrate solution and drawing it across the paper in one smooth motion--no pushing it around--and cutting off the edge where the silver nitrate first contacts the paper. John Coffer seals two sheets together back to back with rubber cement, fully immerses the paper in both the albumen and silver nitrate solutions, and then cuts off the sealed edges.