Learn Russian: Водитель сидит за рулём. - The driver is sitting behind the steering wheel.

The preposition за takes the Instrumental case when describing a static location (answering "where?"), like sitting "behind" the wheel. If the driver were getting into the seat behind the wheel, showing motion or direction, it would take the Accusative case (за руль).

Is за рулём the standard way to say someone is driving?

Yes! While it literally means "behind the steering wheel," за рулём is the most common idiom used to mean "driving" or "at the wheel." If someone calls your phone while you are driving, you would just say Я за рулём (I am driving).

Since руль ends in a soft sign (-ь), how can I tell its gender from this sentence?

You can tell руль is masculine because its Instrumental ending is -ём (рулём). If it were a feminine soft-sign noun, the Instrumental ending would be -ью (like ночью or дверью). Spotting these case endings is a great way to memorize noun genders!

How does the word водитель (driver) connect to Russian verbs of motion?

It is built directly from the multidirectional verb водить, which means "to lead" or "to drive." By adding the suffix -тель (which shows who does an action, similar to "-er" in English), you get водитель - the person who drives.