Can I throw away my exes stuff?

Send her certified mail giving her 10 days then if no response discard it. Use of a lawyer is recommended.

The best thing to do is to send her a certified letter saying that you will dispose of her items of personal property on a date at least a month away if she does not retrieve the items, in my opinion.

Generally, your question is one of whether or not a bailment has arisen, and you don't provide enough information to be able to answer your question specifically. In general, you should contact your ex in writing (using a method that you can prove later, like sending a letter by US mail, return receipt requested, or at least by email or text message if you can prove that your ex owns that email address or telephone number), and advise her to come on a mutually convenient date and time to get her property, including the dog, or they will be deemed abandoned after 30 days of her receipt of your communication (letter/email/text message), and will be disposed of. If you are confident that you can prove she received your communication, then you may generally proceed to dispose of the property. You should continue donating personal property and clothing, rather than just throwing it out, and finding a good home for the dog. You should consult with an attorney to discuss the specifics of your situation, and your options.

I would suggest you send her a 30 day notice to pick up her stuff and retrieve her dog. In your notice, also state that if she fails to retrieve her stuff and her dog, you will throw everything away and give away her dog. Keep a copy of your letter and mail the original to her by Certified Return Receipt Requested mail. That way, if she sues you for her stuff and dog at a later date, you will be able to prove in Court that you sent her reasonable notice. Good luck.