Is a dog killer a dog murderer?

No. This sounds to me like you speak Spanish or at least that English is your second language.

In English when you kill a regular person, you are known as a killer. HOWEVER, when you kill a politician (like a president or a figure of respect & authority) you are a murderer.

That's why the person who assassinated President Kennedy is a murderer/assasin but the Chapman dude that killed John Lennon (from the band The Beatles) is a killer & is refered to as John Lennon's Killer. It works the same way with assassinate & kill. You kill a normal person you assasinate a president.