Is there any good proof yet, that 1-2g fish oil per day helps prevent headaches, helps get over anxiety symptoms, or lessen depression feelings?

Hype. Krill oil and fish oil are the new snake oil, both are high in omega 3 fatty acids which are extremely fragile. Omega 3 fish oils reduce triglycerides by impairing the liver's ablility to make them. However the oxidative stress caused by fish oil actually raises LDL cholesterol. Some day we may use the phrase "this is just more fish /krill oil" or "here comes the fish/krill oil salesman".

Some. The research is just getting started. One study has shown benefits for depression and anxiety. I am unaware of any studies showing benefit with headache.

Yes, there are. This is an excellent review from the univ of md on the benefits of omega 3 http://www.Umm.Edu/altmed/articles/omega-3-000316.Htm.