Do all sugars have an anomeric carbon? How can we tell on a sugar other than glucose?
Any sugar that can cyclize by opening up a carbonyl group will have that carbon as anomeric. A very small sugar (triose) cannot cyclize (too small) despite the presence of a carbonyl, such as the aldehyde on glyceraldehyde (consult the index of your text to see the structure of glyceraldehyde).

Do anomeric C have to be carbonyl, because you defined in your lecture that an anomeric C is a cabonyl C.
The anomeric C of sugars starts out as a carbonyl in the straight chain form.

What is the difference between an anomer and an isomer?
Any molecule with an asymmetric carbon can form a stereoisomer (sometimes referred to as an optical isomer). Sugars that can form those special isomers that put a hydroxyl either above or below the hydrogen sharing that carbon are anomers of each other. Note that the carbon involved in anomer formation is asymmetric (has 4 different groups attached) after cyclization but was not asymmetric before cyclization, when it was a carbonyl carbon.

I am under the impression that a polysaccharide can only be formed from the anomeric carbon - why is that so? Can't the C 2 or 3 also react? And can the anomeric carbon react with C 2 or 3 on a different sugar?
One anomeric carbon mustbe involved in the formation of a glycosidic bond; the partner can be any of the non-anomeric (or anomeric) carbons. The anomeric carbon probably has to undergo a partial ring opening during the formation of the glycosidic bond (while retaining its alpha or beta designation).