Correlating certain metrics with Sessions and Prod

I have a requirement in the GA api to get a few very specific types of metrics:

Site Level

Site Visits with a Cart Add: This is a count of how many sessions occurred for a time frame where a cart add occurred in the session.
Site Visits with a Product View: This is a count of how many sessions occurred for a time frame where a product detail view occurred in the session.

Product Level

Product Visits: This is how many sessions occurred from the scope of a product. For example, sku 12345 had 200 sessions touch on that product in a time frame. If it helps, Adobe Analytics has this built in as a "visits" metric that can be correlated per product.
Cart Visits: This is how many cart opens (deduped cart adds) occurred from the scope of a product. This would be directly correlated to Adobe's "carts" metrics that can be correlated per product.

From all the research I've done, and many tests, these four items don't appear to be possible just using metric and dimension correlations. I've heard that they might be possible using some sort of segment filter configuration, but I'm not sure how those would even begin to look.

I'm wondering if anyone has dealt with this type of thing before and would be able to help point me in the right direction. A couple json examples would be even more helpful, or a couple screenshots of how these segments might be setup in the interface.

Thanks!

Edit 5/1/17: I was able to figure out the two site level metrics above by applying the segments against sessions:

For the Cart Adds - sessions::condition::ga:productAddsToCart!=0
For the Product Views - sessions::condition::ga:productDetailViews!=0

I feel like this could solve the problem for the product level metrics above, but I still can't find any way to get session data on a per-product basis.

Edit 5/4/17: I figured I'd attach a picture for the product visits on what I am trying to accomplish. This is on the query explorer:

And this is the results I get, instead of the expected list of skus - each with a number of sessions:

Circling around - it is clear from GA that they don't support correlating certain values, because the scope is different. The only two options you have to getting something similar is to:

Use a custom dimension/metric to achieve the correlations you need, especially since these can be given the specific scopes you may need.
Change your requirements/expectations for the business rules.

In the case of the application I was attempting, the latter made more sense for the users of the application.