Taxes for American Retiree in France

I have tried to research this, and I am wondering if someone can confirm or deny these impressions.

1. If I live in France less than 6 months per year, I don't pay any income tax on my US pension (but would pay in the US of course).
2. I can buy a rental place with a guaranteed income for 10 years - and possibly can find one, where I can spend a month there a year?
3. If I was to live in France more full time, then a salary of say $80,000. It seems my income tax would be 30-40% in France, but maybe less than 20% in the US alone.
4. Are there other countries nearby (e.g. Germany or Italy) with more favorable terms where one could visit France?
5. More generally, where can I get these kinds of questions answered? Come to France and sit down with a French accountant perhaps?

thank you in advance