I have managed to reproduce the sitemap.ashx problem on the demo site. If you look at the https://demo.mojoportal.com/sitemap.ashx output as an anonymous user now you will see:
however the first page cannot be viewed by anonymous users. The second page can.
(assuming someone else doesn't muck around with these pages before you get to them).
I don't think this is a fully cached output, as new-page-3 appeared in the output as soon as I added the page and refreshed the sitemap output.
The way I achieved this was to change the view permissions of new-page-2 from being visible to all to being visible only to admins. However that may be a red herring - I now find that on creating a new child page of /new-page-1, the new page (/new-page-4) appears in the sitemap immediately even though it is not visible to anonymous users (as it inherited the permissions of /new-page-1). Yet /new-page-1 is not in the sitemap. This may be the important point - the pages where this is going wrong are child pages of a page that is not visible to anonymous users, and not in the sitemap.
I hope this gives enough for you to track it down.
On the other issue, I cannot persuade the last mod date visible in page settings to change on the demo site when I make edits to the page (e.g. add html feature) or to html content on the page - it only seems to change when I edit the page properties themselves.