We spent the night outside

"We spend" would need to be "we spent" as this is the past simple of "to spend". So it would read: "Due to an earthquake, we spent the night outside". "In the open" would be a more common alternative to "outside" giving you: "Due to an earthquake, we spent the night in the open". To 'go out at night' implies going out for the purpose of entertainment so you would not say "We were out all night" in the context of an earthquake. Hope that helps!

The sentence are correct yes. The only thing you must keep in mind is what you people to know. For example. We stayed out (all night.) the key word in that sentence is, "all night" We spend the night out. this can mean you went out for 2,3,4 hours last night, it does not specify how long . "We were out all night" is the same as "We stayed out all night"