Changing important page URLs in redesigns/website updates is highly unprofitable. Links to that page/url will be broken and that does harm SEO efforts. My suggestion would be to keep the same URLs as the old website (whatever those may be). If the old website used .htm, then use .htm as the file extension. You should be able to use crustless links if the old file names were index.htm for every page.
There is no true difference in .htm and .html files, except for the obvious lack of l in one. The most important thing here is that you don’t break any links to your website/content, as those are extremely valuable in the world of SEO.
Be sure to resubmit your website/sitemap to Google and search engines again!
Also, see one of my replies to a topic on SEO for one page sites - most of it applies to websites in general.
Hope that makes sense.