That's quite well possible. Likely ws2811 or something similar.
If the strip automatically starts generating a pattern when powered up, then yeah, probably. You could remove it and plug in your won device on the data line(s).
Take a look at what Adafruit calls NeoPixels. You can control them with an Arduino and you can program them to do just about anything you can imagine!
That MIGHT be exactly what you have but the controller probably isn't an Arduino or anything you can re-program.
In general, I don't recommend buying this kind of thing from eBay, AliExpress, unknown 3rd-party Amazon suppliers, etc. I buy from actual electronics suppliers that have the manufacturer's part numbers and full specs, or links to the manufacturer's datasheet.
...eBay might be fine if you didn't want to modify the thing.
There are various configurations of addressable LED strips. Usually the controller ("brain") is separate. Then there is a "smart" driver that receives the data, takes care of the color & brightness, and then passes the data along to the next driver in the string. Usually that driver is built-into the LED so you don't see a separate chip.
12V LED strips usually have a driver for every 3 LEDs. The voltage is "shared" and 3 LEDs in a row are addressed as one and they have the same color & brightness.
"NeoPixels" use a serial addressing scheme that only requires one data line from the controller. There are only 3 connections to the strip - Power, ground, and data.
There is another type that has an adds a "clock" connection (4 wires/connections). It's less common and the software is simpler.
Yes, exactly. Especially what driver/led controller chip is used. There's a couple common types and they don't all talk the same language. WS2811/12 is common, but not the only possible candidate .
Yes, you can control those from an Arduino. And yes, you need to supply them with 12V or 24V, whichever your product requires. Ensure that the GND of this external power supply is connected to the GND on the Arduino.
I've never heard of a strip that's 12V and 24V... As I said above, 12V strips are usually addressed in groups of 3, and I'll add that 24V strips are usually addressed in groups of 6.
You can probably see the chip so you can tell how many LEDs are controlled together and there should be associated "cut" marks (where it's OK to cut the strip) and that will also tell you how many are addressed together.
