A hotdog or a hamburger

1) I am sure he went to that restaurant and ate a hotdog or a hamburger.

2) I am sure he went to that restaurant and ate either a hotdog or a hamburger.

Would '1' and '2' be considered false if he ate both a hotdog and a hamburger?

Is the 'or' necessarily exclusive?

3) I am sure he went to that restaurant and ate a hotdog or drank a soda.

4) I am sure he went to that restaurant and either ate a hotdog or drank a soda.

Would '3' and '4' be considered false if he bot ate a hotdog and drank a soda?

Is the 'or' necessarily exclusive?

Gratefully,

Navi.