No, you should NOT say "He sticks them onto his bedroom's walls."
It should be "He sticks them onto his bedroom walls."
Here, bedroom is used as an ADJECTIVE - it modifies the noun "walls". You can use words like bedroom, kitchen to describe other nouns. See the examples below:
"He purchased a brand new kitchen sink."
"I looked out my bedroom window."
"I think the living room walls need repainting."
"I am tired of looking at the same old kitchen wallpaper"
"The courthouse ceiling is magnificent!"