NO, you CANNOT import html code into EverWeb - that is what people have been trying to tell you - EverWeb and the likes of other programmes like it have no import facility, so CANNOT import html code of any sort, which is why you have to cut and paste.
If you want to import code, then you will have to purchase DreamWeaver from Adobe or Flux 6 - both of these have import facilities, but your site will probably not look the same as iWeb html is loaded and it really won't be worth doing.
As for Google website builder, this is designed to be used with Google.
If you want to re-do your site, give EverWeb a try.
Updates and changes in iWeb: There won't be any further updates to iWeb - it is unsupported software. As for updates to OS X. Yes, these will definitely happen and then at some point in time iWeb will stop working, so you really need to think about moving on now rather than later, before you are forced to.
Time to move on - there is better software out there now and other programmes can do a lot more than iWeb ever could - in the days when I used iWeb, I still had to use html code to get the kind of site that I wanted.