I see that some royal icing recipes call for cream of tartar and some don't. Not sure if that makes it stronger or what? If I am making something like a spider web that needs to be very strong should I use cream of tartar or not?
Yes, or you can use a little vinegar also.
i use royal all the time and its strong enough with just icing sugar and powdered egg white. of course it is a brittle icing and wont take a knock but it is self supporting. hth
When I need to pipe anything really thin I stir the icing a bit. This releases some of the air and makes things less brittle.