When your device is floating, if you then connected any single pin of it to "actual" earth, that rail of it would match the potential of "actual" earth, and all other elements of the evice would maintain their relative voltage diferences internally. Remember, a voltage is meaningless alone, voltages must be between points. For a floating device lets say you have a 5V, 3.3V and Gnd rail. Now if you connect 3.3V to "actual" earth, the Gnd rail stays 3.3V below 3.3V to become 3.3V below "actual" earth, and the 5V rail ends up as 5-3.3=1.7V above "actual" earth, relative voltages unchanged and without spiking, device operation unaffected.
And you don't seem to even be so much wiring it to "actual" earth via a lowresistance connection, you just happen, to my understanding, to be touching the device as a human being who may or may not be at some potentialrelative to "actul" earth. To my understanding that is the only contact possible for this device, contact by a large fleshy resistance with some capacitivative elements to it. You might consider ESD protection TVS diodes, perhaps, if you often touch multiple separate points within the device, but for an initial prototype this will almost definitely survive long term without them so long as you aren't constantly charging yourself up by pacing back and forth on carpet in a low humidity climate and then touching, simulataneously, separate points on the circuit. I'm not an ESD expert, but if you start a thread specifically about "what ESD protection do I need on a button I'll often touch", I'm sure you'll get detailed responses, they'll mostly involve special fast acting zener diodes and perhaps some extra resistors to keep the "sensitive"* MCU pins safe from what happens at the button itself.
*An arduino nano is not like a really sensitive MOSET or analogue device, it will cope with normal handling just fine most of the time, ESD protection here would be for improved long term reliability and adding some extra certainty, it isn't a matter so vital to this application that you can't start anything without addresing it
Running off a transformer based wall wart is electrically no different to a battery operated device. Just make sue you get a wall wart which does use a transformer, most good quality ones should.
As for capacitance, this is about decoupling caps. A usual rule is a 10u or 1u cermaic in parallel between the power rails (5V and Gnd) of each IC, physically as close to the IC as possible.
As for current consumption, a glance at the display's webpage shows it can be powered through a small connector, I doubt a large current would be delivered through that. I haven't looked it to all its details, but I'd be very surprised if at peak consumption it could possible draw >750mA. I strongly suspect 1A will be perfectly ample for that display, an arduino and some buttons.