Sure. Syria is no friend to the US. The Assad regime is strong allies with Iran who as you know has an adversarial relationship with us. Getting Assad out, weakens Iran. I wouldn't consider weapons in the hands of Assad "stable" as they are known weapons trading partners with places like Iran, North Korea, and Russia. As more democracies are established in the middle east, we put more pressure on the autocratic regimes in places like Iran and diminish the power of the anti US zealots.
This is not an ideology, it is a practical issue. There is no reason our domestic policies should be the same as our foreign ones but last I checked, the vast majority of the government believes citizens should be able to own guns so that's not really an issue anyways.
I see no reason to.