"For the past three years, they have lived with us" is a more commonly used phrase
"They have lived with us since 2017."
"They have lived with us for 3 years."
"They came to live with us 3 years ago."
"They have lived with us since 3 years ago."
"They have lived with us since 3 years."
SINCE is used with a gixed date or time. FOR defines a period.
Here it could be Ria's sentence or they came to live with us 3 years ago if you want ago. Since would need 'since December 2015'
I agree with you guys, this sentence sounds wrong to me too.
"have lived... since" or "lived... for"
The combination of "since" and "ago" in the same expression is not good English.