For some reason pins 0 and 1, the rx and tx ones? don't like the LEDs they just flash
They are used as you say for TX and RX so they can't be used for LEDs and communicating with the computer at the same time. You have to override them in software so best not to bother.
and pin 13 is quite uncooperative with fading them.
Not all pins will output PWM and so not all can be used in fading.
says the 4051 can't deal with brightness?
It's the only one of that list that does deal with analogue signals, all the other are on and off only, which is what you need if you are using PWM to fade the LEDs.
Please read the the articles on the playground on multiplexing and PWM.