Understanding 3rd Normal Form Violation in Database Normalization

Yes, it's in 1NF.

You can't side-step the often hard work of determining all the candidate keys by hanging a number off the end of the table and saying, "There. I've got a primary key." One natural candidate key for this table is {Name, Bought from, Date bought}. Consider using "Time bought" instead of "Date bought".

Your definition of 2NF is wrong. Instead of

Second Normal Form: A relation that is in First Normal Form and every non-primary-key attribute is fully functionally dependent on the primary key.

you need something more like this.

Second Normal Form: a relation that is in First Normal Form, and every non-prime attribute is fully functionally dependent on every candidate key.

The term non-prime attribute doesn't mean quite what non-primary-key attribute means.

Your definition of 3NF is wrong, and it's wrong for the same reasons as your definition for 2NF was wrong.

Instead of this

Third Normal Form: A relation that is in First and Second Normal Form and in which no non-primary-key attribute is transitively dependent on the primary key.

you need something closer to this.

Third Normal Form: A relation that is in Second Normal Form and in which every non-prime attribute is nontransitively dependent on every candidate key. (There isn't a really good way to express all those negative in one sentence.)

You should note that not every country has the same rules for addresses. I would believe that address_id (as an integer) is just a surrogate key for an entry in the Address table. It is not a reference to the country.

If you want to look into the variations in addresses you might find the following interesting.

FRANK'S COMPULSIVE GUIDE TO POSTAL ADDRESSES

Regarding City/State/Country you will not always have all of these values available to you. What about countries that do not have states?

If you have a comprehensive data source for every city, state (including NULL cities and states), and country that is maintained, then you could avoid the extra column.

However, you will need to determine your workaround when some elements are missing. (As you can find examples in Frank's web page.)

My personal opinion (not shared by everyone) is that a state and country column are often better than maintaining a large data set.

If you need is extensive enough you could subscribe to a service to maintain you address information. Just do a web search to find a service that fits your budget.