Your tap handle has a mid point between far left and far right ,at which it will blend hot and cold water together.
The tap tails below the sink are connected incorrectly ,for you to get hot on the left and cold on the right they need to be swapped over on the supply pipes.
The cartridge is...
Only one pic visible. It sounds like the tap body needs to be turned on the sink ,and your hot and cold connections to the tap are the wrong way around.
Hot should be lever to the left and cold to the right. The indice on the tap is showing the opposite ,but that can be pulled out and repositioned
That's clearly intended to have the shaft connected to the fitting and not suspended on the flex.
Before fitting to the ceiling ,the shaft gets pulled down ,connects to fitting ,secured by grub screws. The excess cable contained in base plate.