The MacMini running iTunes should be fine.
The AppleTV supports 720p and 1080p and several other resolutions apart from interlaced ones such as 1080i, 576i.
Your TV should be ok, but AppleTv has traditionally been fussy with HDMI. HDCP is required for protected content, some older sets may fall down on this, so hard to say it will definitely work - check the manual.
Home Sharing allows you to browse your itunes library via AppleTVs Computers icon. You do not drag and drop content to AppleTV as it has no storage, so your itunes will need to be running. If you have iTunes store purchased movies/tv shows you should be able to play them from 'the cloud' without the Mac Mini running at all provided your internet connection is reasonably good and not throttled/capped.
'Home made video' provided it has been encoded correctly for AppleTV will play. If not you will need to re-encode which is time consuming.
UK iTunes Store stuff works fine for me in the UK.
It's copy protection over HDMI, some old sets don't fully support it. It's part of the HDMI specification not a different port.