I think when you press the remote button and no response, is because the door is not closed properly. I don't like this as well. For Myvi, if there is a door didn't close property, when u press the remote button, it will lock the doors, but alarm will go off in seconds, telling u that the doors are no closed properly.
For Axia, it just gives no response. So every time need to make sure see the signal lights blink only know the car is locked.
If the alarm is engaged and you open with key, the alarm won't go off immediately, there is a 5 second delay according to the SA. After that the alarm will sound.
There is no reset button for Axia alarm, just plug in the key are turn to ON will disable the alarm. The key must have the correct immobiliser chip otherwise the car won't start.
I have the same problem like you last time, took me few hours to figure out, it is just too tight for first time use. For my case, the seat is adjusted to the lowest position, so I can't press it down anymore. Try pull the adjuster up with a little bit force, eventually you will feel the seat going up, let go the adjuster and pull again to go higher.
Hope this helps you

It is trolling me when I lock, it considered as properly closed and allowing me to lock, but when I try to unlock it realized I didn't shut it properly and disable the remote, locked me outside...