Configurator only works when the configration.h was created with the config tool. If not you can still use config tool, but you need to fill in values by hand looking up from the config file you have. You always see the define name next to value name in config tool to help you.
If I understand you right the display is the only part not working? In thingiverse config file they selected type 11 which is the RepRapDiscount Graphic Controller - the probably most widely used display if you include the compatible clones. I have no idea why they also included ui.cpp - pins are defined in DisplayList.h for displays. And the init seems correct:
#ifdef U8GLIB_ST7920
u8g_InitSPI(&u8g, &u8g_dev_st7920_128x64_sw_spi, UI_DISPLAY_D4_PIN, UI_DISPLAY_ENABLE_PIN, UI_DISPLAY_RS_PIN, U8G_PIN_NONE, U8G_PIN_NONE);
#endif
that display uses the ST7920 chipset.