Simple enough. Here's the image:
OK, so
everything about this is wrong!
Let's start with some basics.
An electrical circuit is a closed loop; it includes both supply and return connections. In electronics, we give one of these connections a special significance, we call it "ground". Why do we do this since the circuit always has two sides and they are of equal importance as both are always required?
Well, the reason is that electronic circuits consist of multiple individual circuits simultaneously serving different functions. One function is to provide operating power to the various devices, while another is to send control signals or "data" from one part to another. The "ground" is in general the common return to many circuits, it becomes a single wire that pairs up with many others to form different functional circuits.
So for starters, whenever we speak of "providing power" to anything, we always mean two wires, "power" and "ground". And in the same way, providing "data" means connecting both data and ground wires. If you are needing to provide both data and power, that means three wires, travelling together.
And the "travelling together" is also important here. Microcontrollers, like all computers, operate at Radio Frequencies ("RF"). A wire that is separate from its partnered ground wire, is called an "antenna" - it will transmit and receive any "RF" signal. We do not want that - we want all of the data generated by the Arduino to go to the LED strip and no other signals to be picked up by the LED strip, so the data wire and its ground reference wire must always be a pair using twin or "figure eight" cable. But since the power wires are carrying fluctuating currents due to the switching of the LEDs (which use PWM internally) and since the ground is also common to the data wire, then the power wires (supply and ground) must also travel as a pair. So in fact, all three must travel as a pair. 
Next, when we say "power every 60 LEDs" (3.5 Amps at full brightness), we mean to power at both ends of the 60 LEDs so the LEDs at each end are equally powered since the flimsy foils on the strips cannot carry much current without dropping voltage. This clearly means you do not cut or break the strip; you have the heavier power cable - preferably 2 mm2 gauge - travelling along with the LED strip (because it is paired with the conductors on the LED strip) and each 60 or so (the figure is not critical) LEDs, the power cable connects across to the 5 V and ground pads which are provided on the strip.
This is of course, somewhat inconvenient but necessary if you expect to operate the strip at anything near full power. Mind you, full power (white) will light up not just the stairway, but the whole room! 
So what do we have here? We have the LED strip(s) in a continuous run. Together with them we have a power cable alongside, and every so often, the 5 V and ground are bridged over to the strip. The power supply may be connected to this cable anywhere; either end or even somewhere in the middle if that was the most practical. At the "data in" end of the strip, the 5 V power and the data connect - three wires together - to the Nano.
And yes, the Nano is far more practical than the UNO, you do not need to disconnect (only) the 5 V wire from the "5V" pin when you connect via USB to a PC as you do with the UNO. It is also cheaper, smaller and you can either solder to a bare Nano board or use header pins to connect to a "solderless breadboard"for prototyping and then stripboard or a "terminal adapter" for the final design. (Or even a custom PCB.) Connecting to the header sockets on the UNO is quite inconvenient unless you use a suitable "shield".
Noting that for the stair lights application, you will want to connect other things such as sensors to control the animation. The "Vin" and "barrel jack" are not useful for power connections, just a novelty for playing with the Arduino alone, or with a few simple LEDs or such.
