GPS to spark core

First of - you’re right Core Vin -> GPS Vin and Core GND -> GPS GND.
For RX/TX you have to be careful to watch that Serial1 RX/TX on the Core are 3.3V and on the GPS RX/TX are 5.0V, so if you connect Core RX to GPS TX, you’ll blow your RX pin, if you don’t do level shifting.
You might be better of with Serial2 (D0/D1), if you don’t want level shifting - although this would be the safer way (e.g. via the Shield shield).

How can you tell it’s not working?
What is the expected behaviour on power up without satellite sync?

The fact that the red LED is not blinking might not be enough to actually tell, that it isn’t working.

As mentioned in the other thread, you will not get a (stable) satellite sync inside a building.

Try to communicate with the module via Serial2 - have a look at the datasheet for commands and expected response.