calmly为方式副词，修饰took，为方式状语。

as if为连词，通常单独引导方式状语从句。但这个句子比较特殊，as if与前面的as calmly 呼应，构成了as...as if...的特殊结构，这种情况比较少见。此时as if并非引导方式状语从句，而是引到比较状语从句，呼应as calmly.

其实as if= as he/it would if

所以，这个句子中，as后省略了he would.

He took the violin from the wall as calmly as if he were a welcome visitor. 

= He took the violin from the wall as calmly as he would if he were a welcome visitor. 

他就像一个受欢迎的访客那样镇静地从墙上取下小提琴。