Thanks for the pics.
1) Looks like a 3 wire cable was installed at the switch instead of a 2 wire cable. In the switch box, it appears that the white wire is not connected? Or is the switch box white wire looped thru the box?
How many wires were originally connected to the dimmer switch? 2 or 3 wires?
2) At the ceiling box, how many house white wires do you have?
3) The only method to confirm if the ceiling remote is working correctly is to have an AC voltmeter available. Perhaps you can borrow a meter from a friend or a neighbor? If not, AC voltmeters can be purchased at any Home Depot or Lowe's for $25 or less.
4) At the receiver output.....ie........receiver side with 3 wires, the black to white (120 volts) with the remote transmitter ON is for the ceiling fan speeds. At the receiver output, blue to white (120 volts) with the remote transmitter ON is for the light kit. Blue to white on the receiver output side needs to be confirmed with a voltmeter.
5) A voltmeter can also confirm if you have 120 volts inside the light sockets.
6) Are the DIP switch settings in the receiver and transmitter matching?