I live in Israel and I have both American and Israeli citizenship. I recently opened a stock investment account in Israel. My questions are:
1. As a dual citizen, do I have to report my market income to both American and Israeli tax authorities?;
2. Do I have to pay tax on this income to both countries?

You DO have to report the Israli-based income on your U.S. tax return.

If you paid taxes to Israel, you can claim a Foreign Tax Credit using Form 1116 on your U.S. tax return.

For the U.S. the rule is clear:
A U.S. citizen or resident must report the world wide income.

Agreed.