Do you know when Adi Sankara lived? Scholars do not even agree on when he roamed the earth. People who have studied Bhashya have told me that based on linguistic analysis (nature of Sanskrit language) that he has not written a lot attributed to him He probably lived BCE almost 1000 years earlier than what the western scholars thinkCould you kindly give evidence that Adi Shankara did not write all stotrams?
As far as I know only Bhajo Govindam was a combined effort with him and his shishyas.
Adi Shankara did install the idol at Mukambika too.
He had no issues for murti worship too.
All stories about him are just mythological stories. We have a member here who argued passionately how Adi Sankara disliked certain grammarians. All these are nothing but imaginations.
The only concrete things available are the Bhashyas and some mantras (like Prarasmarana stotram) with Vedantic content, I am repeating what I have heard and do not have first hand knowledge of these. But I trust the scholarship of the sources.
Adi Shankara did not install any Murthis. Murthi worship was pervasive much later than his time in earth. Idol is not a good word to use since it means satan for Christian and Islamic faith.
I noticed that Mr Prasad has given specific examples to your question