Does the HTML export retain all animations, transitions, and sounds, ect... Is there anything that is lost?
No, everything should be there if you export it to HTML. I tried it before. Not sure about how it will work with third party fonts though, but it should work. Sounds, animations will all be retained.
Is the HTML export responsive?
I used that feature pretty often. It is pretty responsive on my 16GB RAM Mid-2012 i5 MBP. Still depends on what hardware you're on and how big is that presentation. Most of the time for me it works perfectly.
Does the HTML export have any display issues when shown on a mobile device?
No, it works pretty fine on my iOS device, though I have to flip it to landscape to be able to view the whole slideshow. I don't think this is optimized for mobile devices so it is kind of buggy when you view it in mobile device.