Lie vs. Lay (grammatical Nemesis of Mine)

Are these correct? I say yes.

He lay there a second ago.
He lies there now.
He laid (PLACED) his head on the pillow.
He lay on the couch yesterday.
The problems that lie ahead...
The problem that lies ahead...
The state LIES (RESTS) south of the border.
I laid (PLACED) my head on the pillow.
NEVER: I lie my head on the pillow.
As I lie there, I become angry. (PRESENT)
As I lay there, I became angry. (PAST)

Thank you.

 

yes!

 

Thank you.

 

One more, please. Thank you.

The mountains lay before us as we proceeded west. (past)

The mountains lie before us as we proceed west. (present)

 

you've nailed it... no more worries [on lay/lie, anyway!]

 

Thank you.