Your questions are so broad that’s it’s impossible to give you specific answers, so here are some broad answers.
1) That depends on what else the micro has to do. Most GPS modules use a serial interface (SPI,I2C,UART) to communicate, so pretty much any micro can handle that.
2) Pick a GPS module that meets your requirements (that you don’t seem to have). There’s nothing in “configuring” it that can make it work “better”.
3) I think you’ll need another layer to translate long/lat to a specific city. For instance, you’ll notice that your smartphone needs to download data from the network in order for your map app to work; it needs more than just GPS.
4) Broadest question of the bunch. For low power, use low power devices. You could just put everything to sleep and only wake for a measurement once a minute or once a day.
5) Would an external antenna improve performance? Yes.
6) I have no idea, and I’m pretty sure the developers are not telling anyone how their app works.