But for the Arduino and the Servo it seems that Vdd/Power and Vss/Ground are the same thing. Any reason not to label them similarly?
I'm not even sure what the second D in Vdd means. I use it because that's what it says on the datasheet, lol. But I understand at least one of the D's is for "drain," and that Vdd is used to denote the supply pin of device that is composed of mostly CMOS/FETs or to denote the logic bus/rail. In my domain this is usually a microcontroller or logic chip. When BJT logic chips were the thing, it was Vcc, not Vdd - presumably one of those C's was for "collector." Perhaps it stands for Voltage Drain-to-Drain, for CMOS logic push and pull output transistors.
So as far as I know, Vdd is the same thing as power/supply, just a little more specific. The only reason I can come up with for not using Vdd/Vss for anything is that someone may come along and correct you, lol. Is there any reason to use Vdd/Vss, at all? Personally, I find it useful because it is very short and unambiguous to write and read, and it sorta designates what voltage is your logic rail. CMOS stuff typically operates at 1.8 to 5V, but depending on your circuit, it could be anything between.
Also is the labels from the power rails and to the ground even necessary; or are they just on there for clarity?
I dunno why I put those on there. Certainly not necessary until it is necessary. And here, it is not necessary.
Yeah, your calculation of base resistor is fine. It's more of a stupidity check. It's not that critical. In this case, I'd use whatever resistor I had between 1K to 2k. Anything from 300R to 4K would probably be fine, depending on your duty cycle, perhaps. The higher you go, you might get less draw through your LEDs. I think maybe the Hfe will show how much the transistor would draw from an ideal power source, e.g., voltage source with zero impedance.* So you would add the impedance of the transistor to that of the LED strip, and you might not get "full power" when you go much higher than 1.1K, and it will dissipate some extra heat in the transistor, which is probably the main failure concern when you use too little base current. But as long as the transistor doesn't get hot at 99% duty cycle and the LED does what you want it to, then you have enough base current.
*But this would depend on the absolute voltage, so I'm missing at least one piece of the jigsaw puzzle as to what Hfe is exactly a measure of. For most of the reasons one would have to know this stuff, the work is already done for us. We have opamps, voltage regulators, and buffer IC's to choose from, galore.
The new schematic looks fine, to me.