Lay means "to place something down flat," while lie means "to be in a flat position on a surface." The key difference is that lay is transitive and requires an object to act upon, and lie is intransitive, describing something moving on its own or already in position.

Is it would lie or would lay?

The difference in the present tense seems pretty straightforward: lay refers to a direct object, and lie does not. ... The past tense of lie is lay, but not because there is any overlap between the two verbs. So when you say, “I lay down for a nap,” you're actually using the verb lie, not lay, despite the way it sounds.

Would lay down or would lie down?

You lie down, but you lay something down. Lie does not require a direct object. Lay requires a direct object. The same rule applies to laying and lying (not lieing—beware of spelling).