I paid an illustrator to do the art for my picture book and I now own the artwork. Am I then allowed to submit to agents as an author/illustrator? How does this work?
No, that would be dishonest. If you say "I am an illustrator", that means that you draw illustrations. The fact that I bought a piece of beef at the grocery store does not make me a cattle rancher. I own the beef, but I didn't raise the cow.
You can say, "I wrote this book and I also own the copyright on the illustrations" or something to that effect.
If you call yourself the "illustrator", you set yourself up that a publisher might say, "hey, can you make one more illustration?" or "would you be interested in illustrating this other book we have?", and then have to admit that you are not an illustrator and embarrass yourself.
Since you do already own these illustrations, you can use them as a self-publisher, or, in the case that a traditional publisher does want both, you should be entitled to all the royalties without splitting them. However, the original illustrator should still be credited. It's at-the-least unethical to claim their work as yours, even if you have legal ownership of it. Also, you should definitely make sure you have all rights and expectations for the images (and the compensation) spelled out in a signed contact.
Contrary to popular belief it is NOT necessarily better or easier to sell a picture book with illustrations attached. If you ARE an author/illustrator, your best bet is to submit a portfolio, establish yourself as a proven illustrator, and then pitch a book where you've also written the text. Author/illustrators are the gold standard of the industry, true, but the usual career path is to master one side of the slash first, then the other.
It probably won't hurt you to submit a manuscript with pictures attached, but it may not help either. If the story is intimately attached to the pictures, you can provide sketches, but don't expect fidelity to them. If you are primarily a writer, don't commission illustrations unless you are self-publishing. At best, your commissioned illustrations will probably serve as a "proof of concept" that will help the editor visualize your final book. Most publishers like to pair new writers with proven illustrators, and vice versa. Furthermore, they can probably afford much better illustrators than you can.
Not a lawyer.
You paid an illustrator to provide artwork for your book. You (supposedly) own the right to publish the artwork - that's what you paid the artist for. But that doesn't make you the illustrator of the work. You are not an author/illustrator. You are an author. The illustrator would need to be credited as such. The book would be "written by X, illustrated by Y". By "owning the artwork" it is usually meant that you have the right to publish it in your book, not that you can stick your name on it.
The details would be in the agreement you signed with the artist. Agreements that say you are allowed to put your name on another person's work do exist, but that would be explicitly stated, hopefully cost a lot more money, and I personally find it morally disgusting.