First, one would establish that there is actually a correlation.

If so, then is there a causation? (eg, does being in a red state make one suicidal, or does being suicidal affect one's political views) That seems a bit of a stretch. But if a causation CAN be established, then is there anything that can be done to help people?

You forgot the third option: are both caused by a third independent variable.

For instance, economic situation could influence both political views and mental health.

Another possible correlation is aging populations. Among males at least, suicide is highest among males over 65. (https://www.nimh.nih.gov/health/statistics/suicide.shtml) The "hypothesis" might be that lack of access to good health care among this group (either due to location -- rural health care is not in the best of shape right now from what I can tell (https://news.aamc.org/patient-care/article/health-disparitie...) -- or economics) might enhance suicide risk.

Access to firearms could be another correlation. Firearms are the leading method of suicide in the United States. The consensus opinion I've heard is that firearms are a lot easier to make an attempted suicide an actual suicide than most other methods.

All of these are hypothesis; I doubt we can come to real good conclusions without better data than these summary pages.

That is true.

There could be even other causes. Living in an area with less dense population affects one's outlook in various ways.