My younger brothers always took the job of picking out what cookie cutters to use very seriously. I'm sure it was the engineers in them coming out at an early age.
Cookie cutters with a lot of small pieces can be a challenge for young kids to use. Wider cookies with less angles are easier to cut, and move, to waiting cookie sheets.
Cookie cutters with narrow sections can also be difficult to move. I use a flour-dusted metal spatula to carefully remove the cut cookie from the cutting board. The metal spatula helps prevent the cookie dough from breaking and ensures the cookie gets to the cookie sheet in one piece. When waiting, uncooked dough rests on clean dish towels.
Plastic cups, jar lids and small tins can also be used as cookie cutters. One year we couldn't find cookie cutters in the shapes we wanted so we used new sand toy shapes. Apply a little butter or shortening before dusting with flour and the cookie dough will get a clean cut.