PDF is a page layout format, better suited for press than ebook readers. Some information, like table of contents, might be included, but other things like what parts are page headers and footers will not be. The features you mention aren't in fact part of PDF in the first place; searching works if the text is text, text to speech might work if the text is ordered sensibly or the software guesses correctly from position. There are some restriction flags that can be set to break printing, copying and text to speech in PDF; IMHO those should not exist.
Figures, tables and mathematic formulae very rarely convert well, as PDF will have their graphical shape rather than their structure. I've seen demonstrations that Pocketbook could
reflow PDF with formulae, and
should soon handle images.
Using Calibre to convert to some other format means you're shifting the burden of making those guesses to Calibre, and evens the playing field for the readers quite a lot. Calibre won't restrict the output document. The upside of doing it this way is that you can tune or fix the conversion more easily than in a reader. There's at least
one more thread on this.
If you choose not to preconvert, however, it becomes more interesting how well the reader deals with PDF. Then you'll want to look for features like margin trimming, text reflow, and perhaps multi-column paging.
I guess the primary lesson is that PDF is a horrible source format.