I misread your question, so ignore my last answer. "I read a lot" could potentially mean either "I read often" or "I read a lot of books", but most of the time it would be interpreted as the former.

In practice, it could mean either. It might be rephrased as "I do a lot of reading". The speaker doesn't specify that the reading material includes books but we might assume or understand books.

"A lot " is an adverb here, so the emphasis is on the verb " to read". It means "much" , "often", "frequently". It's not about the number of books you read.

I agree with the other native speakers. It could mean either.

"A lot" modifies "books", not "read". It refers to the quantity of books that you read, not how much or how often you read. So it means "我读很多的书" instead of "我经常读书".