Do the email triggers respect product inventory?

Yes, emails that recommend specific purchases to be made, such as the abandoned cart, wishlist reminder, and product recommendation emails only include products that are currently in stock.

The stock is queried at the time that the email is sent out. There isn't a delay in the sync process.

Certain emails that don't depend on inventory such as review requests and purchase anniversary emails don't look at inventory.

By default, the stock status table is checked to see that the products are In Stock.  You can also require that a positive stock quantity is found in the Advanced settings: