I am just a Web Site owner hosted by LunarPages and I am in no way associated in part, or in whole with LunarPages but I will try and answer your questions.

LunarPages hosts all kind of sites from business, eCommerce, forum/posting sites, news sites - you name it, it's probably hosted by LunarPages.

There aren't very many restrictions here unless something you are doing uses up a heck of a lot of RAM, CPU or Bandwith resources - basically anything that will deminish the enjoyment of other people whom have a web site on LunarPages.  

I can't see why fan sites would not be permitted and as for hot linking your images from eBay and what not.. yup, you can do that but always monitor how much bandwith you are using.  There are limits - 20 gigabytes with the standard plan and 60 with the premium one.  If you have a very popular, and rather large image that everyone in hot linking too, then your bandwith can be eaten up in no time flat.  

I run eBay auctions and use my own web site to house the images.  No problems there.

Hope this helps.  Remember, advice from LunarPages may differ but these are my observations since coming aboard about a week or so ago.

Quite a lot of hosts don't allow hotlinking, because people just abuse the services.

Here at lunarpages, almost anything is allowed.