Specifically referring to the “double” contents, that looks to me like a “page header”, being added by the book reader itself. Does that text not change from one page to the next as you go through major sections?
It’s often a good idea to verify that kind of stuff by looking at the actual source itself, to make sure what you’re looking at isn’t something the reader is adding or changing. Compiling to ePub and opening in Sigil is usually the best way of doing this, but if you want to see exactly what is going into the Mobi output, then go into the general options tab of the compile overview screen, and tick the checkbox to “Save source files…”. You can look at these files directly in an HTML editor and verify whether “Contents” is in fact really printed into the output twice.
(As an aside: Amazon recommends using ePub with Kindle Previewer now, not Mobi.)
I’ve added this section back into the manual. It was fixed for ebook output it looks like, and should all be good to go at this point. The only setting relevant to the contents page is the Page title style. If you’re using the built-in Ebook format you shouldn’t have to do anything here. But if you were creating your own from scratch, you would want to create a style in the Styles pane, that establishes the look for the heading, and then wire it up in this pane. This setting also impacts the “Notes” section, if you make use of endnotes in your ebook.
As to the general efficacy of the compile settings upgrade process: basically if Scrivener 1 supported the feature it should map over if you use the legacy format it creates for you. The upgrade guide that Katherine referred to above goes over how to audit that to your specifications.
It’s not going to create stuff that didn’t exist in v1. As noted there were no styles (and the styles used in the compiler are separate from the project anyway), and you couldn’t create a custom table of contents page either, so there certainly are things you would need to add to an older project, if you want to take advantage of new features.
I’ve always advocated working the other way around: bring the settings you want into the project, rather than trying to take an empty template and getting all of your content over into that (that said, if you really do want to do that, there are far better methods than manually going through with copy and paste!). The user manual has a little guide for doing things either way, in §5.4.2, Converting a Project to a Different Template, pg 56. For what you need, points (2) and (4) will be the main things you want to look into.
That said, the challenge listed above, for example of not having a “Table of Contents” section type, is relatively easy to fix if you just want to add these feature to your existing project.