I got a whole leg of lamb, removed all the big parts of meat for another dish, then scraped down the remaining pieces around the bones and the fat. Should I use all of it for stock or does the pure fat parts for instance make the stock too greasy?
Stock is made from bones only and broth is the liquid that meat has been simmered in. A well made stock should be clear without particles or cloudiness. Broth will usually be somewhat cloudy due to containing more dissolved proteins.
Bones for lamb stock (and any brown stock - veal can be made white or brown but lamb is usually brown) are typically first roasted. The roasting will melt fat that may still be intact. After roasting bones for stocks such as lamb, veal, and duck you would then remove the bones from the roasting pan and roast the vegetables. You can use the residual fat for coating the vegetables prior to roasting. (Just put vegetables in roasting pan and toss to coat with the fat). Fat from the bones will congeal on the top once the stock is chilled and you can then easily remove it.
I wouldn't add or purposely leave chunks of fat attached to the bone as you just have to remove it later but what is there isn't a problem.
Re chicken stock vs. broth = this is in addition to the other great explanations. When well chilled do not skim off congealed fat on top of either chicken stock or broth because that is where the flavor is - without that it will taste like warm dishwater. But Yes, always skim off fat layer on beef, ham etc. broth & stock or all you will taste is the grease.
When I saute boneless chicken breasts, or roast bone-in chicken breasts, I always deglaze the pan and pop that great liquid in a dated freezer bag and stash it in the freezer. Then when I want to make chicken & dumplings or chicken soup I bring out all my little bags full of great flavored frozen gold!
If your animal is a lamb, you can use any part of it. However to answer the question in the headline, some animals (of which the moose is the most likely to be cooked) have poisonous levels of vitamin A in their livers, and you should therefore not use their livers for any culinary purpose.