If your floor was finished in the last few decades, it is likely that the finish is polyurethane. Do not wax polyurethane! There are a number of cleaners made for polyurethane floors that may remove some surface material that dulls a finish, but does not leave a residue.
Large area - If the finish in the large areas is dulled from abrasion, the only treatment that will restore the overall shine level is screening the whole room (or at least a large defined area up to an edge where a slight shift in gloss will not offend).
This is a process of slight abrasion using a commercial buffing machine (rented from a home center) that takes off the gloss of the whole floor. It is much less intense than a complete sanding and refinishing. The floor is then cleaned thoroughly and refinished with a new topcoat of polyurethane.
Paint globs - The paint globs must be chipped up. You could try a flexible bladed putty knife slid under the edge. If that is not successful, scraping carefully and gently on the top of the paint blob until you just get down to the finish, sometimes the thin last layer will flake away from the finish without much damage.
If the finish is damaged under the blobs, a light sanding followed by a feathered brushing of new polyurethane (of the same gloss as the floor) should cure it. It might be a bit shinier that the surrounding area, but it will dull a bit over time, or you could dull it down after it is well cured with a gently rub of an abrasive dish pad.
Stair edges - The stair edges should be sanded overall, onto the stair tread by about 1". You can use masking tape to make the sanding and subsequent refinishing straight and even. Use several coats of new finish on the edge. When dry, remove the tape. The transition from the old finish to the new on a stair edge will not be very noticeable. Again, you can use a mildly abrasive pad to soften the transition.
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".
Best Answer
YES
You will need to lightly sand the surface and thoroughly clean it, then add one or more additional coats of a compatible product that has the level of gloss you prefer.