A single XML sitemap should not contain a mix of HTTP and HTTPS URLs (i.e. essentially different locations as far as the search engines are concerned). So, the sitemap located at
http://example.com/Sitemap.xml should only contain URLs starting
http://example.com/ and similar for the HTTPS sitemap.
Q: My site has both "http" and "https" versions of URLs. Do I need to list both?
No. Please list only one version of a URL in your Sitemaps. Including multiple versions of URLs may result in incomplete crawling of your site.
From GWT Sitemap Errors:
URL not allowed
https://www.example.com/ — it's using a different protocol (https rather than http)
So, you would seem to need 2 sitemaps. One for HTTPS and the other for HTTP. Although if you only have 1 URL in your HTTPS sitemap (your homepage!) then whether you need an HTTPS sitemap at all is debatable - although it could help with reporting in GWT.
To answer your specific questions...
(1) Should the sitemap be located at HTTPS and is there any problem if we:
Well, yes and no. You would need a sitemap at both, but primarily at HTTP.
(2) include the HTTPS version of homepage (not including the HTTP homepage)
Include the HTTPS version of the homepage in the HTTPS sitemap. You state you have a redirect to the HTTPS homepage, so there is no HTTP homepage. So, the HTTP homepage should not appear in any sitemap.
(3) include the HTTP of all other pages within the HTTPS sitemap
No. You would need a separate HTTP sitemap.