Just a few math answers to common renovation questions.
How far should steps go up (rise) and how wide should steps be (run)?
The rise should not be less than 4 inches or you will stumble on it, nor more than 9 inches or it becomes a ladder. Then within those limits, the rise times the run should equal 75. There are several other formulas that are valid, but this one works well for standard stairways.
How long should a nail be?
Theoretically the nail is three times as long as the board you are nailing through. That will put twice an much nail into the receiving piece as through the piece you are nailing on.
Nailing 3/4 inch plywood to a joist would require 3 x 3/4 or a nail 2-1/4 inches long. 3/8 inch molding would work best with about 9/8 inch nails -- so either 1 inch or 1-1/4 will work fine.
When the formula doesn't work
When you get to larger wood the formula breaks down because a 4-1/2 inch nail for nailing for standard 2x framing would split the wood, but for 3/4 inch boards and down, the formula works pretty well.
How much of a space should be left between the window frame and the rough opening of the window?
1/2 inch all around to leave room for levelling and shimming.