Hello Tuomo,
This an interesting situation. You can partly solve Copper Pour issue by changing Flood Over parameter to Thermal Connections, however doing this will not solve the problem of the trace coming to the center of Via or a Pad.
I suggest to change the pad stack to be Annual ring style and print just the Pad/Vias on a clear film and tape it to board prior to drilling and use it as your guide.
Regards, Yan