If one were to use a word like 'sycophant' repeatedly, would that cause others to think he was smart?

Not necessarily..could look like the person was just trying to impress others.

It would make me believe that the person had just learned the word and was trying to impress people with his vocabulary.

No. It makes him look like he only knows one semi large word.

Only if you used a menagerie of different words to accompany it. If you use a word too often people would start question the intelligence of said individual.

Only if he were actually talking about a sycophant, and that's pretty boring in itself...

No.