For example, if I draw a 10cm diameter circle in Adobe Illustrator, export it as a DXF, import it into Lazycam, then into gcode and into mach3, will the circle at the end actually be 10cm diameter?
Provided that Illustrator outputs the .dxf properly. From what I've read, Illustrator's .dxf export is not very good at all.
One thing to note. You need to use the same unit of measure throughout the process. From design to machine.
.dxf files typically won't be drawn with any specific unit. What I mean, is that your 10cm circle will really be 10 units. So, If you draw a 10cm circle, and I draw a 10 inch circle, and someone else draws a 10mm circle, all 3 .dxf files could be exactly the same.
You machine will most likely be set up in mm's, so, typically, you would draw the circle with a 100mm diameter. Some programs that export .dxf files have a tendency to assume your drawing in a specific unit, and will output in another unit, by actually scaling the drawing when exporting. You'll need to do some testing to make sure the .dxf's you export are the size you expect them to be.
Since my machine is set up in inches, for me to get a 10cm circle, I'd draw it with a diameter of 3.937 inches.
Steps/mm can be easily calculated. Tell us how it's driven, and the specifics of the components.
Or, use the calibrate function in Mach3. I think it's on the settings page.