Its not wrong for a non muslim to say salam to a muslim. Muslims have to reply to that. Albeit if a muslim wanna greet a non muslim, its preferable to use other phrase even though the ruling isn't haram to say salam.

If you are very particular with language, then do this. In the sense of linguistic, due to the "Al / As" referring to a Proper Noun which in this case As-Salam refers to the name of God, then omit the As / Al. Then you will have "peace be upon you" and not "Peace be upon you"

Although for me I don't see anything wrong with using As-Salam as a Proper Noun, because when I say that to a non-muslim, I can think of it as "may God be with you and open up your heart to Islam".

Thats just my 2cents though.

then if we can use it, we say it salamualaikum, right?