Correct on the spark plugs.

The starter spinning and not engaging is a sticky or bad starter drive. Time for new starter unless you can locate a new drive ("bendix") and replace it. If you have a good auto electric shop nearby they can probably locate the part or rebuild the starter. If you do take the starter apart it is a good time to lightly oil the bushings, check the brushes for wear, etc.

The starter is not causing the engine to run poorly unless you have major electrical problems.

Greg