Well you know more about Psoc than me. I've only been using PICs so far.
1. depends on the project requirements. An ADC takes an amount of time to perform the conversion so the question is how frequently do you need to take readings? Can you get the required bandwidth out of just one ADC? If not then you will need three ADCs.
Another consideration is: Can you spare three pins for three ADCs? If you can't then that settles the question for you.
On the other hand what is easier? Does it make programming easier or harder to use three ADCs or to multiplex three inputs into one ADC?
2. No idea. But this question may be rendered moot by the answers to 1.
3. Yes. Unless the psoc has 3 CPUs on board you won't be doing anything simultaneously; you will read from one ADC, then the next, then the 3rd. When you display stuff on screen just display all three values in different places.
If psoc has a multitasking OS and you're just writing a thread to read one ADC and display it on screen, and having 3 of those threads, then you'll have to serialise LCD access (i.e synchronise your threads so they don't try to write to it at the same time) otherwise you will confuse it. Also if you MUX the ADCs you'll have to serialise that as well, so it may just be easier to do the whole lot in one thread. You wouldn't be looking at a vast amount of code anyway.