What is the proper procedure to apply multiple coats of Polyurethane?
When you apply multiple coats of urethane, any dust or small flaws will be amplified with each new coat. If you want the smoothest finish possible, then wait until it is dry, very lightly sand with very fine paper (220 or finer) to remove any dust bumps, wipe it clean with a slightly damp cloth or tact rag, let it dry or buff it dry with a dry cloth, then apply the next coat. There is no reason to use mineral spirits, actually I would advise you not to use mineral spirits.

I see no advantage to applying successive coats with, then against the grain. As long as you apply an even coat with a good quality brush or foam brush and don't overwork it, any brush marks should disappear. The finished product should be so smooth that you couldn't tell which way you brushed the urethane on.  Also, if you are using a satin finish urethane, be sure to mix it well, stirred, not shaken. The dulling agents tend to fall to the bottom, so it is important to mix it well. No matter how many coats you apply, it should not become "glossy".