When a non-muslim dies

Is that merely an assumption? Queer, you're arguing about something you dont have enough knowledge.

"salaam alaykum" was NOT the customary way to greet in pre-Islamic Arabia. People use to wish each other with words like "an'im sabahan" *and other words derived from "*An-Ni'mah" which means good living.

Ibn Qayyim states in his Bada'i' Al Fawaid (144)

[QUOTE]
“Allah, the Sovereign, the Most Holy, the Peace, prescribed that the greeting among the people of Islam should be ‘al-salaamu ‘alaikum’, which is better than all the greetings of other nations which include impossible ideas or lies, such as saying, ‘May you live for a thousand years,’ or things that are not accurate, such as ‘An’im Sabahan (Good morning),’ or actions that are not right, such as prostrating in greeting.....
[/QUOTE]

Yes because this was what the Prophet (sas) and his companions (rah)practiced. The actions of the companions, which are documented and preserved in the books of hadith and seerah, serve as evidences for the correct interpretation of the ahadith.

It'd help if you approach this Deen with a pure intention and with a desire to learn knowledge and seek the right path. Any other intention would only result in a lost cause....because this Deen is DIVINE and what that means is that you cannot play around it nor prove it deficient in any way or form!