Okay, several things going on here.
First, FXL/fixed page EPUB is a problematic format on many levels. It's a "works or it doesn't" process, and if there are flaws, they are difficult to resolve. This is compounded when the export is from a developed print layout. The assumption is often that FXL is the obvious or simple choice for e-book export, like export to PDF, but the truth is that it has all the hurdles and limitations of reflowable EPUB with many more of its own. Unlike PDF, which was developed to emulate a printer, EPUB needs an absolutely meticulous document structure to work with, including hidden elements and layout methods, or it will export with flaws. Put simply, it's a poor choice to rely on and if your first attempt doesn't come out well, it can be difficult to put right because there are probably sloppy "hey, it works in print!" aspects to the source document.
Second, EPUB and Kindle are not the same format at all. There's no reason to expect any given EPUB document, reflowable or FXL, to convert to Kindle without changes and flaws. In developing books for all platforms, a print layout must be specifically adapted to an export for optimum results on generic EPUB, the common Apple/iBooks version of EPUB, and Kindle.
Next, EPUB supports only three image formats, JPEG, PNG and GIF. InDesign converts any other format to one of those, and can be set to convert all images to one of them as well. If the export isn't good enough, and adjusting the few export options available doesn't fix things, the only good solution is to replace all source image files with one of the supported formats, with strong preference given to JPEG over PNG. (It is a nearly absolute rule to never use or convert to GIF, which is an obsolete and cranky format for anything but animated forum avatars.)
So those are the areas causing problems for you — you'll have to work at the source file until it gives the desired results for both FXL EPUB (using a "vanilla" EPUB reader like Calibre as your proofing tool) and Kindle Previewer. Besides needing two separate export files, know that Kindle does not handle fixed-page books very well or on all devices, making it an even poorer choice for that platform.
Happy to help further if the above doesn't get your project sorted out.
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