I’ve tried two different setups for Hangul input.

The first combination (GNOME+X11+IBus) just worked—it was trivial to setup from the GNOME Control Center and there were no issues that I could find. Last time I tried this was around a year ago but it had been working for years before that.

The second combination (Plasma+Wayland+IBus) barely worked: I could enter characters but couldn’t add space or other symbols between characters, and settings were all over the place and would randomly stop working. Last tested ten minutes ago.

I’ve heard people getting better results on Plasma+Wayland+Fcitx5 but it’s not something I’ve tried.

Do languages that use non-Latin alphabets (Asian, Cyrillic, Greek, Hebrew) have upper and lower case letters?

Greek has upper and lower case. From mathematics/physics you may have come across e.g. the ones for sigma (Σ, σ). Cyrillic also has them; most look the same between the upper & lower case variants, just bigger/smaller (Л, л), but there are some that differ (А, а).

I don’t think most Asian scripts have letter cases. Javanese script does have upper case but only for a small subset of letters and they are generally not used anymore.

What about serif or sans-serif?

Cyrillic and Greek, yes. There are also equivalents to the serif and sans-serif typefaces in Chinese, Japanese, and Korean typography.

How do they show emphasis?

On the Web, boldface (but not italics) is very commonly used across various writing systems. Obviously no all caps for those without capital letters.