Learn Arabic (Modern Standard): هذا مقعد السائق.

In an Idafa (possessive) phrase like مقعد السائق, the first word (the thing owned) never takes the prefix ال (the). It is already considered definite because it belongs to a specific person. Only the second word (the owner) takes ال.

How do we know the phrase means the driver's seat and not a driver's seat?

In an Idafa, if the second word (the owner) is definite because it has ال, the entire phrase becomes definite. Since السائق means the driver, the whole phrase translates to the driver's seat. If it were just سائق without ال, it would mean a driver's seat.

Where is the word for is in this sentence?

Arabic doesn't use a verb for to be in present tense sentences. You simply put the subject هذا (this) next to the predicate مقعد السائق (the driver's seat). The word is is naturally understood!

If the driver were a woman, would we change هذا to the feminine هذه?

No, you would still use هذا. The word this points to the seat (مقعد), which is a masculine noun. In an Idafa phrase, the gender of the first word (the thing owned) controls the grammar of the sentence, regardless of the owner's gender.