Is it possible to actually upload a picture (.svg) into this and make a curved image?  I've tried and tried and keep getting error messages.  Does "picture" just mean text or can I upload an image?

I'm working with my students to make custom Christmas presents.

Try putting your image into windows 3d Builder and turning that into an STL or OBJ to modify in Tinkercad.

I thought they had a curved text utility though? Either way, windows 3d builder is a surprisingly powerful tool and should get you an importable 3d object.

3D buider is probably the best bet if you want flexibility.   Tinkercad does have a curved text shape generator but the options (font) are limited.  

In 3D Builder,  under edit use the "emboss" option.  You will immediately have access to a field for text entry,  projection type and font selection.   3D builder lets you write .STL files & you can print directly from there if you want.

In addition to direct entry of text, the emboss function allows you to load a pattern & emboss that on a surface.  This would probably be the way to import your calligraphy & embed it on a ring.