Invalid code does not equate to slow code. Google used to purposely not close html and body tags to save bandwidth and loading time. I took the question as meaning w3c valid code.

As long as links are not broken and resources are not returning 404 errors, my understanding is that it is not affected at all. The only thing I would make sure of is that fetch as google returns the page looking like it should. Here is a video from the authority on the matter.

https://www.youtube.com/watch?v=j3KgrbiB1pc#t=52

How many products are you going to have? Depending on your url structure, you might think about using the word buy instead of product or shop, so your urls could be buy/fancy-health-product.

I wouldn't for the cart and checkout, I would actually no-index them. The reason being is depending on what measure you are using for cart abandonment, having people land on those pages will skew your results. Also at the same time there is no good reason for them to enter on those pages so it will increase the overall bounce rate if people are entering on them.

Privacy policy, shipping, and return policy, I would not specifically no-index them, but I would not try to optimize them for anything other business name - page name. That being said, if you are like Costco and offer some amazing return policy that someone can return a product after 10 years for a full refund and it is a popular product; I might consider optimizing the return policy for that as well. Or if you are the only person that offers free international shipping on a large item like a baby grand piano or something. But if you fall in line with the rest of the industry I would just leave them as is and only optimize them for your business name - policy type. So then if people search for your business and return policy those pages will more than likely come up.

Yes, there is a schema for brand. It does need to be inside a product though from my understanding. You can find out more information about how to use it here, https://support.google.com/webmasters/answer/146750?hl=en  Also note you can nest organizational info into the brand name as well, so not only can you have brand you can information about the brand.

Does the platform you are using let you select a default category for the product? Several platforms will let you select multiple categories and make you chose one as a default category. If so, you can just work off the uri of the page and insert the default category as the canonical category in the rel=canonical.

Google Webmaster Tools has that built in. That is the only one I am familiar with.