12-0-12 means you can get TWO 12VAC outputs frm the transformer. If you said it's rated at 5A, then each output can deliver 5A. So, each output power is 12 x 5 = 60VA (not watt as it depends on the power factor of the load). So, the total power can be drawn frm the outputs is 120VA. If you connect it as 24V output, you still can draw 5A, as the total power is still 120VA.
Thickness of the secondary wire is similar to that of primary doesn't mean it can handle the similar current rating that can be handled by primary. This is because the number of wire turn for secondary winding might be more and therefore the winding resistance is higher.