1.) You should never use those square/rectangle +9v batteries.. they suck. (notoriously known for lack of current)
only when you need to.. I mean they arent gonna blow anything up.. but most likely wont last long. (power through USB when testing)
2.) Yes.. if the Arduino you using is a +5v based version.. then most likely you will just calculate the resistance for +5v. ** There are +3.3v versions out there of Pro-Minis.. Nanos too maybe? other variants as well **
* So if you had the 'accent led' (like yours above) you would probably have one leg (GND) in the GND I/O pin or breadboard rail.. and the other leg (V+) in the I/O pin on the Arduino you want to use to control it.. so you would a resistor on one side of the connection. (I dont believe it matters what side)
Then whenever you turn that I/O pin on, on the Arduino HIGH, the LED will turn on.
Since you are *powering it from the Arduino it self, you would use that to calculate the resistor value.
* The Arduino has very low limits on what it can 'power/drive' directly from itself.
Each I/O pin on the board can only output a MAX of 40mA (some say 20mA)..
With I think a 'combined' current output of 200mA (or close)
So really your small accent leds are really the only thing an Arduino can directly power (and other small current need devices of course..but most modules/chips exceed this limit)
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Neopixels:
You are more or less correct. They are (can be) RGB leds (you really should google this term to get a visual aid on all the things people do with them).. that usually are in a strip, but can be purchased individually, as well as in other patterns/layouts (rings/stars/sticks...etc) each of these leds can be its own color and brightness.. so you can fade from white to red to green to purple....rainbow...whatever you can code it do.
If you using more than a few, you would need to power them externally, like from the battery source itself.. or something (they have a STRICT requirement of +5v).. but usually need way more current than an Arduino pin can provide.
The key element here is that can be 'daisy chained' together.. with the Arduino only using 1 I/O pin to control, 1-1000's of LEDs..
All the same 'kit' (effect).. but shows a fire effect you can do..
Another Arduino based project: (just a fun general project, that is Arduino based)