This is a constant struggle for me and I’ve found there’s no way around it other than test, test, test…
For each course I create, I test across a bunch of browsers (IE11, Chrome, Firefox, Safari) and go through the entire course to ensure everything is working correctly.
I’ve also found that some live environments (web server via FTP as well as LMS) often disallow certain filetypes which can play havoc with embedded media in courses.
The rendering issues you describe (elements not appearing correctly) usually only happen for me in older versions of IE – however, local IT policy can be set up to force IE11 to display in Compatibility View, meaning that while you’re using IE11 it could be displaying content in IE10/9/8/7/5 mode which don’t support a lot of elements! Worth checking with your IT department or LMS vendor to see if this is the case.