[DiapDealer]

reading in Kindle app on iPad

I can't verify what the issues with Kindle for iPad might be, but with regard to the Kindle device itself:

The front cover image is repeated on the second page

I've never experienced that issue so I can't help much

There is no navigational TOC

Not exactly sure what you mean, but the toc.ncx file will only be used to create chapter "jump points" that can be navigated using the Kindle's five-way controller. They'll be represented as dots on the progress bar. If there is no in-line (html with the "toc" semantic assigned) toc, kindlegen does not create one from the toc.ncx.

Justified text doesn't hyphenate
Headings (aligned left in CSS) do hyphenate

The Kindle doesn't hyphenate. Period. There will be occasional ragged-right lines in justified text.

CSS working properly? Does Kindle treat H1, H2 etc in a default fashion? Or does it use the CSS? If the former, am I best to style all headings with P tags in order for Kindle to use my CSS?

Some CSS works fine, some... not so much. Use KindlePreviewer (just drag & drop the epub on it) to test. You can also use MobiUnpack.py to see what html kindlegen produces. The biggest pain in the ass is that kindlegen doesn't support multiple CSS classes assigned to elements. So:

<p class="bigger bolder>Blah, blah, blah</p>

ain't gonna fly. You'd have to do something like:

<p class="bigger"><span class="bolder">Blah, blah, blah</span></p>

With kindlegen, the less CSS—and the simpler the CSS—the better. Complicated doesn't work. It all has to be flattened into html 3.2 attributes inside the mobi (no CSS in mobi) by the converter program.