ATMEGA32U4-AURCT vs ATMEGA32U4-AUR vs ATMEGA32U4-AU?
I am not sure where you have got the part numbers that are in your question title, but it looks like for a ATmega32U4 you have only four options (see page 422): two package and two oscillators.

The -AU/-MU select the 44ML/44PW package respectively, whereas the 32U4/32U4RC select the oscillator option. They both should be the same internally except the default oscillator. They both need an external oscillator if you want to run them at > 8MHz. See page 29 section 6.2.1 and 6.2.2 - the difference is how the fuses are programmed when shipped. So, the 32U4-AU is set by default to work with the external oscillator, whereas the 32U4RC-AU is set to work with the internal oscillator and both are 44ML TFQP package. So, if you get the 32U4-AU, you will need to connect it to an oscillator to be able to reprogram it.

As for the additional -R letter I would guess that it signifies that the parts are packaged and shipped in a reel.