First let's start with the Census records.
In 1900 William and Anna C. Warnecke were living in Salmon, Lemhi Co., ID.
William Warnecke, b. Germany May 1865, boarding beer.
Anna C., wife, b. WI, June 1862, had given birth to four children but only two alive in 1900.
James A. Thompson, boarder.
In 1910 the Warnecke family was living in Butte Ward 8, Silver Bow Co., Montana.
William D. Warnecke, b. Germany, 44, M1=10 yrs, saloon keeper.
Ella C., wife, b. WI, 47, M2-10 yrs, gave birth to three children & all alive in 1910.
Arthur W., son, b. ID, 8, father born=Germany & mother born=WI [obviously the Warnecke's].
Kate Cosgrove, lodger.
In 1920 the family was living in Lake Avoca, Silver Bow Co., MT.
William H. Warnecke, b. Germany, 54.
Anna, wife, b. WI, 57.
Arthur, foster son, b. ID, 18, single, miner copper mill, father born=Sweden, mother born=Utah [probably his biological parents].
In 1930 the family was still living in Lake Avoca.
William Warnecke, b. Germany, 62,florist flower shop.
Anna C., wife, b. WI, 65.
Arthur, son, b. ID, 28, single, both parents born Sweden [probably biological parents].
William, Anna and Arthur's obituaries might provide more information but I'm not familiar how to find death dates in Montana.
The BIG question is was Arthur legally adopted? He might not have been but the place to start is to look for his birth certificate. If he was legally adopted, you should find it under Warnecke. If he was not legally adopted, you'll need his birth surname to find it.
I'd inquire about his birth cert at the Lemhi County Clerk-Auditor-Recorder's Office, lemhicountyidaho.org/clerkauditors.htm . Most birth certs aren't available until 100 years after birth but it's been over 100 years.
If they don't have it, he might have been legally adopted and you'll have to go through the courts.
"All inquiries for adoption information should be directed to the county of origin or to the Center for Vital Statistics. . ." I can't give you the URL for this quote because they changed the web site and I can't find it at the new web site. Successful Searching!