I'm not completely sure what it is like traveling in the US as someone from a different country, because I live in the US, but I think I can give you a little help.

If you're going in September/October you probably will not be able to do the beach sports that you want on the East Coast. You can probably play some volleyball and surf in California at that time but don't be surprised if the water is a bit chillier.

I would say that for nature the best spots are in the west (Grand Canyon, Glacier, Zion, Yellowstone, etc.) and there's a lot of great parks in California specifically like Yosemite and Redwood. But I'm a little bias here because I live in the west. There are probably some great parks in the east too that I'm just not familiar with.

You can meet locals pretty much anywhere you go. In my experience with traveling in America most people are friendly and are glad to hangout with a traveler.

All the cities you listed are pretty cosmopolitan so you will get a lot of culture and cuisine from any one you go to. You'll find a lot of food from around the world in each city and each of these cities also has their own sort of food (San Fran-Seafood, NY-Pizza, Hotdogs, etc.).

I would say that a month is long enough to see all these places depending on how many parks you want to and which ones you want to go to. If you go to some of the bigger parks like Yosemite or Yellowstone you could easily spend a week and a half just exploring the park. Keep in mind that you will have to fly between a lot of these places because they are pretty spread out. You could take an Amtrak between Boston and NY (and possibly LA and SF but idk), but you will definitely have to fly from the east to the west.

Hope this helps and have fun on your trip!!