Everyone is using verbs and prepositional phrases, but you can also put "rather than" in front of other parts of speech. Rather than a dog, the animal was a cat. Rather than red, her coat was yellow. Rather than slowly, she walked quickly. Rather than inside, his house is outside the forest.

Yes, it is grammatical and natural to start a sentence with "Rather than, " such as your example sentence. If someone really wanted to quibble, they could criticize your sentence for lack of parallelism -- meaning that the second half uses "by" but the first half does not. A more parallel sentence would be either: "Rather than by taking exams, . . .. by doing essay assignments" or "Rather than taking exams, doing essay assignments helps students learn more effectively." But in my opinion at least, your sentence is perfectly good despite lack of parallelism.

Rather than going out tonight, I'm going to stay home and study

Thanks Robert!

Yes you can. In your example it should be changed to "by taking exams". Rather than spending all day on the internet, let's go for a walk in the forest.