Hello and welcome to JustAnswer!
P1684 sets when the transmission control module has lost it's battery feed at some point. This one is most likely nothing to worry about and shouldn't be related to your no start problem unless the battery is actually very low.
It wouldn't hurt to try a jump start if you haven't already but I don't believe that's going to be the issue.
It sounds as if the problem is with the aftermarket alarm. It would be extremely helpful if I knew what alarm was on the car and I could then track down a manual. Each alarm has different connections and functions.
The problem could be with the car itself - ignition switch, broken ignition switch actuator pin, range sensor, starter, starter cable, starter relay, etc - but the fact that you don't hear a chirp when turning the key on and it seems you did before makes me believe that the problem is alarm related. Most alarms will interrupt power to the starter in one way or another, usually being tied into the ignition switch wiring. It's possible that the switch or actuator pin could be bad and the alarm module and car don't know you're turning the key to the crank position. You can use a test light to check for power on the yellow wire at the ignition switch with the key in the crank position to rule this out.
If that's ok then there's a very good chance this is alarm related. Your choices at that point are going to be to either have the system removed, or if you'd rather keep it then you'll have to identify what it is and go from there. If it's something you'd want to keep and know what the system is then you can let me know and I'll attempt to find a manual for it.