Subfolder all the way. There have been numerous tests done on this question, and they all show that the subfolder option gives you the benefit of the main domain. When you create a subdomain, search engines look at that as a whole other site, and you do not get the same benefit from being associated with the main domain.

There is no benefit to hosting the main site on one hosting configuration, and the separate subfolder site on another hosting configuration. I think I know why you are asking this (related to C-blocks), but I would advise hosting both the main site and subfolder site on the same hosting, as Google will look at it as part of the same site anyway. Kudos for going for the subfolder route - you lose some SEO benefit when you go the subdomain route.

A lot of the SEO jargon is not going to make sense to most business owners (and will only make sense to your internal marketing team, if you have one in place).

That said, you can judge how well your SEO consultant is doing by three metrics:

1) Are you getting more traffic?
2) Is that increased traffic resulting in more sales or other goals?
3) Are you moving up in the rankings?

The proof is in the pudding, and if these three things aren't happening, the rest is inconsequential.

But, the way that your SEO consultant goes about getting results DOES matter. You want to be sure that the steps they are taking will have long-lasting effects that don't disappear the next time Google cracks down on "black hat" SEO techniques that seek to manipulate the rules.

The two things your SEO expert will be doing to help your rankings improve will be On Page Factors and Off Page Factors.

On Page Factors will be easy to spot. They may add or improve content on your site, change the layout of your pages, tweak the design, change the URL structure, or organize your content better.

Off Page Factors mostly comes down to link building. Ideally, your SEO consultant is building links in places where your buyers already hang out. In other words, you would want these links, even if search engines didn't exist, because they would drive qualified leads.

What you want to avoid is hiring a black hat SEO who builds back links to your site on Private Blog Networks (Google this term) or in other shady locations. You may be happy in the short-term if your ranking improve, but black hat link building can come back to bite you in the a$$ and even earn you a Google penalty.

To sum up, you'll know if your SEO expert is doing a good job if your phone starts ringing more often and you start getting more online orders.

Be aware that Bing seems to reflect changes more quickly than Google does. For whatever reason, Google seems to implement changes to rankings a bit more slowly than Bing.

A reasonable time frame to see results in most categories is about six months, Depending on how competitive your vertical is, it may be less or more time than this to get the exact results you are looking for, but six months should be more than enough time to determine whether you are seeing progress or not.

Hope that helps shed some light on your question.

If your site is not mobile-friendly, HTTPS, and at least as fast as your competition, don't expect to rank.
Focus on matching searcher intent to the keywords you are trying to rank for. Take notice if all the top results for a given search phrase are informational results or shopping results. Adjust the pages on your site accordingly.
Getting back links from topically relevant sites is still important, no matter what anyone says. Not just any old back links will suffice. I would be looking for back links to your site that you would seek, even if search engines did not exist.