Learn Turkish: Bu vizyoner yazar herkese ilham veriyor. - This visionary writer inspires everyone.

The compound verb ilham vermek (to inspire, literally "to give inspiration") requires the dative case. The inspiration is being given to someone. So, herkes (everyone) takes the dative suffix -e to become herkese.

How could I add the lesson's word hatta (even / in fact) to escalate this sentence?

You could use it to build on a previous thought: O harika biri, hatta bu vizyoner yazar herkese ilham veriyor. (He is a great person; in fact, this visionary writer inspires everyone.) You can also use it to escalate the group being inspired: Bize, hatta herkese ilham veriyor. (He inspires us, even everyone.)

The word yazar (writer) looks exactly like a verb. Is it?

Good eye! It is formed from the verb yazmak (to write) plus the aorist (simple present) suffix -ar. While yazar can act as a verb meaning "he/she writes", it is extremely common as a standalone noun meaning "writer" or "author" (literally "one who writes").

Is there a single-word verb for "to inspire" instead of using ilham veriyor?

Not really. Turkish almost always expresses "to inspire" using the compound phrase ilham vermek (literally "to give inspiration"). Because it uses the verb vermek (to give), you must always use the dative direction suffix on the person receiving the inspiration.