What is the difference between「取り除く」and standalone 「除く」? In the case of similar phrase like 「取り戻す」and「戻す」, it seems that the difference is clearer: there is something about "taking back what was yours" in the former, while the latter can be just returning something to its original state, among other things.
Is there a common theme among verbs that has 取り appended before the verb?
This Chiebukuro post explains the difference between the two quite well. To summarize, 取り除く implies that the subject is (physically) directly taking action to get rid of the object. 除く is often used when the object is abstract and cannot be directly touched. Here are the examples given for what is valid and what is not:
たまったホコリを取り除く○
たまったホコリを除く✖️
To get rid of dust.
これらの条件からこの条件を除く○
これらの条件からこの条件を取り除く✖️
To get rid of this term from a list of terms.
However, I think it is better to look at these words from the perspective of 取る rather than 除く. Usually with compound verbs like 取り除く, 取る is the main action while the appended auxiliary verb (除く) modifies it. 取る, meaning take, usually means that there is a direct action done by the speaker to "take" the object.
取る = to take
取り除く = to take and remove
取り返す = to take back
取り上げる = to pick up
取り出す = to take out.
It isn't 取り that is being appended to the beginning of other verbs, but rather other verbs added to the end of 取り. Other examples of compound verbs include:
繰り返す = to repeat
走り出す = to start running
愛し合う = to love each other
