Overall, your design looks pretty sound. Keep in mind that I have a limited understanding of your subject matter, and only a vague idea of the way you intend to use this data, aside from the query challenge you posted.
There are a couple of places where the design looks strange to me. In particular, the table you have called "NAMES" looks to me like it has one entry for each author. I would have called that table AUTHOR. And I would have called the table you called AUTHOR something like WRITTEN_BY. It's really a junction table between books and authors. But if your choice of table names makes more sense to you, then keep what makes sense to you.
Also, it looks like NAMES is missing a field called entity_id that would reference an entry in ENTITY. Compare with the CUSTOMER table. I'm not sure what name_id is doing in the ENTITY table. It looks like the relationship is backwards, but I'm not sure.
When you go to actually build the database, you will need to add features specific to sqlite. That's not my DBMS, so I'll stay out of that. I will say that you'll do well to declare a primary key for each table. In some cases, this will have to be a compound key, as in your AUTHOR table. You'll also do well to declare any keys that reference other tables as foreign keys. It's well worth the overhead in order to keep you from mismanaging your data.
I'll note in passing that your design of the table ENTITY is an instance of a superclass/subclass pattern and a design technique called class table inheritance. ENTITY (describing contacts) is a superclass, and CUSTOMER and NAMES are both subclasses. You've done a good job here, but knowing the buzzwords may help you find additional reading material.
I had to make a couple of assumptions in order to make sense out of the query challenge you posed. First, I assume that "most profitable" means the one with the most dollar (or other currency) volume of sales. Second, I assume that in the unlikely event of a tie for first place, you'll want customer lists for all the top authors.
Your query is unnecessarily complex, but even if it were ultimately simplified it would still be a fairly complex deal. There are three problems in one here. First is to figure out the dollar volume of the most profitable author. Second, is to figure out a list of NAMES that meet this dollar volume, and third is to join those NAMES and AUTHORS and BOOKS, and ORDER_ITEMS and ORDER_TRANSACTIONS and CUSTOMERS and ENTITY, so as to pickup the contact info for the customers. If I have enough time, I'll take a shot at it, but I don't have the facilities here to try to debug my effort.
Last what is simple to the reader may not be simple to the query optimizer. I don't know how good the sqlite optimizer is at finding an optimal strategy. For some other database products, the difference between an optimal strategy and a mediocre one can be a hundred to one in terms of performance. There are things a database designer can do to affect the optimizer. But in general, it's better to let the optimizer do the optimizing.