How to efficiently manage sitemap.xml for a moderate size website?

There are few concerns regarding this:

1. Where to put the file? Is it generated per request (as an API)? Is it saved on a storage server?

2. When to update the file? Is it every time new content added? Is batch update better?

3. How to update lastmod? Is it that necessary to be precised?

Total URLs I'm thinking is about 10000 existing URLs and possibly 50 URLs daily. Is there anything else to consider for sitemap.xml?