Better known as lanolin, this oil is naturally produced by wool bearing animals. It helps to keep them clean and dry! It is commonly refined down and used in skin care products, lip balms and yes...your gum. Often the term they use for it is "gum base". Mind you, gum base is a very broad term and can be made from other substances too including glycerin, polyethylene, polyvinyl acetate, petroleum wax, stearic acid, and latexin in any combination. There is no requirement in Canadian labeling law that they have to indicate the source on the package. Don't worry though. It hasn't been found to be BAAAAAAAd for you :)
Carmine or carminic acid is distilled from boiling crushed cochineal beetles. It produces this brilliant red hue that is used as 'natural color'. It did come from all natural bugs after all and it is certainly better for you than many of the synthetic dyes on the market.
Beaver Castoreum. I'm not explaining this one in detail here as I already did a post on it. If you want to know what beaver anal glands are doing in your food click here.
Cellulose is a broad term you will see on the side of many packages. It has grown in popularity because it is an easy way to add fiber to products, acts as an anti-caking agent and thickening agent in processed foods. Cellulose can come from a variety of plants, but one of the most cost effective sources is cotton and wood chips. Common uses are in high fibre cereals, dried Parmesan cheese and SMART products (smart breads, smart pastas etc.). Just like the 'gum base' the source does not need to be indicated on the package.