I agree with this to a certain degree. Page views and user behavior tell Google everything they need to know. No one at Google is manually looking at your site unless you are doing something horribly wrong.

A large amount of page views could signal to the Google bot that the site is popular. Page views combined with long on-site time and low exit rates can tell the bot that the page is not only popular, but also very well put together. (engaging)

Chances are, that due to the different copy that surrounds your listing (URL structure, surrounding copy, additional navigation, etc) across the different ecom platforms you will not see any negative effects. I will add that due to the popularity of Amazon and Ebay, your product, if searched for, will appear in search results on there instead of your main website.

Ideally, I would use the different platforms to try and rank for additional variations of your product keyword phrases. Switch up the wording slightly to include more information, or similar information for each listing.

It's a grey area to be sure. Lots of the things that Google states can contradict one another, such as in your case: They want you to have authoritative backlinks from reliable sources, but they dont want you to pay for it. (Might get in their way of getting your Adwords dollars, lol)

In this case, look closely at what you are getting. Sites like YP, Chamber of Commerce all offer paid profile creations, with the paid profile links being of a higher visibility within their website. If you are getting a full profile page, with lots of ways for you to support your business or company then it could be beneficial. If you are getting a small, otherwise unfindable profile page with a anchor text optimized link directly to your site, I would stay away.

Think of the benefits of having the link. Is your link going to be placed somewhere it can be found, and when found, does the page that will be linking offer the user anything. How does the profile page help build upon your brand?

"If I were to pay for a profile with them, would I request for a "nofollow" link back to my site? "
Again, it depends on how they set up the profile, and how they set up their profiles for all the other businesses on that site. If Google sees the site as just a way to milk money out of people for paid backlinks, they will get hit and eventually so will you. I would do some investigating into the other businesses that have profiles on there, and see how they do in search results. Either way, one link will most likely not do a ton of damage to your reputation, but is that a risk you would be willing to take? Just boils down to what you feel comfortable with.

It seems like you are actually asking 2 questions in one. Let's approach them both separately.

Should you include index.html or not in your structure? Personally, I think the cleanest URL structure works the best. Get rid of any URL parameters that do not offer any benefit. Cleaner URLs not only look better, but are easier to read and share. Who really wants to share DOMAIN.com/fishing.index.html vs DOMAIN.com/fishing. It may seem petty, but in the grander scheme of things it will just work better. If you do this in one area of your site, keep it consistant through the whole site

As to making sure that Google only indexes one version, this can be done through the URL parameter redirects on your server. You can create rules where the URL automatically strips out any additional URL items, and redirects to the proper version. Once you create your sitemap, make sure to only include the versions of your URL you want indexed. One thing I have found in my entire SEO career is this: The easier and clearer you paint a picture for Google about your site, the better your results will be.

"I want to build a review widget to display our reviews on our contractors' websites."

If you are going to be displaying it as a widget on their website, I'm not sure what duplicate issue you are expecting. You would not have the same reviews across mutiple sites if you are adding contractor-specific reviews to a contractor-specific site. Unless you are referring to having the review content displayed on your siet where they sign up, and also on their site as a widget. Even if that is the case, you will have so many differences between the two pages and surrounding supporting text I doubt you would see any if at all negative effects.

Duplicate content penalties are aimed at people who use multiple pages of exact or highly similar content to try and manipulate search engine rankings. Just by having a few reviews that are the same across a few pages I dont think you would be grouped into that.

The most accurate way is to do it manually. Doing a site:YOURDOMAIN.com in Google will alert you to most of the ugly URLs you want to get rid of.

If you are moving it to have them host the domain, you should not see any issues. I thought you were speaking about having a .weebly domain, in which case you could see some negative effects from that. Out of curiosity, is there a reason you are wanting to move it there?