I have Moon+ Reader Pro and all the ebooks (25+) have imported fine. The covers come in fine, IF the cover is properly specified in the EPUB file. (Since I'm writing my own software to generate EPUB files from MultiMarkdown I had to learn a lot about it. It's free software if you want a copy.) However if you are searching for the covers online (because it's not part of the EPUB) there is no guarantee Moon Reader will find it.
Go to
http://validator.idpf.org/ and upload your EPUB files that do not have cover art. If it doesn't pass the test, it's not a valid EPUB.
As a highly experienced computer programmer I say you should keep this in mind: Most publishers do no know what they are doing when they convert books to EPUB. They are likely not hiring computer people to do the conversion as I've come across 5+ EPUB books (where the paper version was by an experienced author) with many errors in spelling.
I also have a theory that some illegal EPUB files are being created by kids from a PDF, using OCR software, and the EPUBs also contain a lot of errors.