1. Yes, it's a 1-1/8" steerer. Most bikes use this standard. Wide choice of stems.
2. There's a huge difference between an 8cm stem and a 10cm stem. If one fits, the other will probably hurt. Don't guess.
3. You need to consider the rise angle as well. There's a very noticeable difference between a 6 degree rise 8cm stem and a 10 degree rise 8cm stem. One, the will be different effective lengths. Two, they will offer different ranges of fit. Again, know what you need. Many stems have their lengths and rise angles imprinted on them in a discreet or less than obvious location. Have you checked?
4. The guidelines in your first post are fine, but they leave out another matter of consideration: Stack height. This is the distance the stem occupies on the steerer tube, and it varies between mfrs. I've personally had stems with 35, 37 & 40mm stack heights; there are pprobably more than that. If your replacement stem has a different stack height than your original, you'll likely need to change out spacers or add some thin 1mm or 0.5mm shims. Otherwise, you may not be able to preload your headset properly.
5. Color is secondary.
6. Your source where you obtained your current broken stem should be a good source for the replacement.