Hi,
1. You can but in that case, you need to specify a product SEF name. There is no way for now to be able to empty both or the system doesn't know if you want to display categories or products when a URL is requested.
2. Hard to say yes or no.
It's only correct if the default menu item on your website is either a HikaShop products listing or a HikaShop categories listing.
Otherwise, the menu item alias must precedes the product alias in the URL as per Joomla's SEF system requirements.
So in such case the canonical URL of the product should be: dog-products/test-product-1
3. Supposing that your menu item is at the root of your menu items tree and that the main category selected in your menu item settings is the category "Dog products", then that URL is "correct" for that "Dog products" category and you can use it in the canonical URL field of that category.
4. The URL will be:
mysite.com/dog-products/category/dog-food-products
And thus you should have dog-products/category/dog-food-products in the canonical URL field.
5. That's correct. And no, Joomla doesn't confuse because that is not the real category URL. The real one is
mysite.com/dog-products/category/dog-products
It's able to understand without the category/dog-products part because that category is selected in the "main category" setting of the menu item.
6. My answers to the previous points already cover the things that are problematic.