Does Troubleshooter Affect Synths?

No it only affects robots. Synths are their own category.

What are synths weak to?

Synth weaknesses in Fallout 4

Although synths are equipped with energy weapons they're weak to energy themselves, being, as they are, mostly circuitry. Their armour can give some high radiation resistance, but strong ballistic damage and pulse explosives work too.

Does General Chao's Revenge work on synths?

"General Chao's Sword" (+50% bonus damage to robots) does NOT affect gen 3 synths due to some bizarre oversight, so it is safe to assume that +damage to humans will affect them.