Hello, my friend.
Do we redirect old URL's to the new product page URL's?
If you keep the product and just changing URL - yes.
Do we redirect old mobile URL's to new mobile URL's or to the desktop equivalent?
So, the whole mobile URL is different? you actually have domain.com/mobile-version-url and domain.com/desktop-version-url, not m.domain.com/all-version-url and domain.com/all-version-url? if first option, then yes, mobile to mobile, desktop to desktop, if second option - then redirect everything to desktop version. However, it does depend on how exactly you're handling mobile version requests. Basically, if it's completely two different static, physically existing pages for mobile and desktop versions, then they need separate redirects.
Do we redirect all old product page URL's containing navigation elements to the new product page URL?
I'm not understanding the question completely. Are you saying if you have a link on page to old url, then should you redirect to new url? If so, then yes, you should change the link.
If we have a category page being added to two different sections how do we determine the right canonical URL? (the URL will be different because the customer paths will be different)
Again, not understanding the question. The same category page will be in two different places? Anyway, canonicalize it to the page, which you prefer to rank.
Do we need to make sure and redirect all old URL's to a new URL? If so, what is the best way to find all of the URL's?
Yes, you do. Otherwise it's not gonna be working (or you gonna have 404s). To determine which pages are not being redirected - use 404 checker - there are whole bunch free ones on the mighty internet.
P.S. I'm a little confused about the way you're doing redirects etc. At least from what I understand from your questions. It seems that in some questions you're actually talking about changing links from pages, not doing redirects. If so, please, clarify. If not, then I need more details (or clearer questions).
Hope this make sense and helps.