can I trade options on every stock?

The capital market regulators set standards for which stocks can have options contracts, and which ones cannot. I noticed that thinly traded stocks rarely have option contracts, and I guess the reason is to discourage short sellers from speculating on them, further driving their price downward.

The answer is no! Options are instruments meant to manage risk exposure on the stock. Some stocks (like penny stocks) have little or no trading volume on the exchange so if almost nobody is trading the stock why will the exchange design an option for it? Think about it!