Does the size of a battery effect the strength of an electromagnet?
No, a higher voltage potential is what effects the strength of an electromagnet.
Certain materials are strongly magnetic, such as nickel, iron, cobalt, and their alloys. Whether or not a particular electromagnet can pick up a particular piece of metal (assuming it in one of those on the list) depends on the size and weight of the metal and the strength of the electromagnet. http://en.wikipedia.org/wiki/Magnetism