UTF-8 – Should UTF-8, 16, etc. Be Used for Everything?

Looks like some european language characters cannot be stored in UTF-8?

Not true. UTF-8 can store any character set without limitations except maybe for Klingon. UTF-8 is your one stop shop for internationalization. If you have problems with characters, they are most likely to be encoding problems, or missing support for that character range in the font you're using to display the data with (Extremely unlikely for a european language character though, but common e.g. when viewing indian sites on an european computer. See also this question)

If a non-western character set can't be rendered, it could be that the user's built in font does not have that range of UTF-8 covered.

Update: Klingon it is indeed not part of official UTF-8:

Some modern invented scripts which have not yet been included in Unicode (e.g., Tengwar) or which do not qualify for inclusion in Unicode due to lack of real-world use (e.g., Klingon) are listed in the ConScript Unicode Registry, along with unofficial but widely-used Private Use Area code assignments.

However, there is a volunteer project that has inofficially assigned code points F8D0-F8FF in the private area to Klingon. Gallery of Klingon characters

East Asian languages typically require less storage in UTF-16 (2 bytes is enough for 99% of East-Asian language characters) than UTF-8 (typically 3 bytes is required).

Of course, for Western lanagues, UTF-8 is usually smaller (1 byte instead of 2). For mixed files like HTML (where there's a lot of markup) it's much of a muchness.

Processing of UTF-16 for user-mode applications is slightly easier than processing UTF-8, because surrogate pairs behave in almost the same way that combining characters behave. So UTF-16 can usually be processed as a fixed-size encoding.