Yes, you don't need to run the trace to the image... you can place a via through the image and connect to the bottom side of the via. But, a via is a connector and around that via there will be no solder mask. Although it is not that visible and probably not make any difference. You could place them all in the same corner to make it symmetrical... You could also place the via right next to your touch button and run a short trace to the image the way I had explained, then run you trace on the bottom side form the via to the connector pin.
Yep, copper is copper, DIY or fab house, it should be the same. You should check you gerber to make sure everything is connected and the solder mask looks the way you want it to.