Key remote trunk lock/unlock button?

The trunk also won't open when pressing the blue trunk key.

The lock and unlock buttons are supposed to lock and unlock all the doors and the liftgate together, that part sounds like it's working. Then the blue button should pop the glass only. From your description it sounds like the liftgate glass actuator may be the problem, when you push the button, it should release the glass and it should pop and raise up. Try this, put your hand on the glass like you are going to open it, now push the blue button, do you hear a click and the glass can be opened? If yes, then you have an issue with the spring-loaded pusher that is supposed to "kick" the glass out of the latch. They get rusty and bind up. Take some spray lube and spray it into the glass latch and open and close the glass a few times to see if it frees up, if it won't free up you will need to access it by removing the inner trim panel on the liftgate. Because you at least hear a click I think it just needs to be lubricated.