The suicide rate obviously has multiple contributing factors, but access to firearms is absolutely one of them. There's multiple studies on this that will come up in a quick web search. In general, access to anything that makes suicide more impulsive increases the suicide rate. I say this as a person who absolutely believes that access to firearms should be the default state for those that want it.

This is a myth. The suicide rate in Japan is lower than the US, and similar to European countries. South Africa, Russia, and Korea have bonkers suicide rates.

Opportunity and desire both contribute to rate. Firearms increase opportunity so more of those with desire will try. Some cultures also give more people the desire so more attempts will be made using other methods. It is not either or.

Apples and oranges. Japan has an insane work culture that leads to burnout and depression, and therefore more suicides. Not sure why Uruguay has such a high suicide rate, but they also have a completely different culture than both America and Japan. The method in which you take your life is less important if you are intent on taking your own life. Guns make it super easy.

Access to firearms increases the rate of suicide. He may have worded it poorly but the point stands. The fact that other countries have worse rates of suicide without firearms notwithstanding, because if they had access to firearms, it would be even higher.