Is what the questioner does—researching the fatwas of trustworthy scholars and conveying them to his relatives in Syria—considered a good deed?

It is permissible for you to convey to people what you have learned of knowledge if you purify your intention for the sake of Allah, and you verify the issues and their reliable sources, and the correctness of your understanding of what you convey. It is not a condition that you be a mujtahid scholar, but rather that you understand what you convey. If you issue a fatwa by conveying a fatwa from a shaykh whom you trust, then state that the fatwa belongs to this shaykh, and it is preferable to present the issue to a shaykh or an advanced student of knowledge.