Just to remember for myself (or anyone else):

The Unicode to LaTeX converter does not handle combining accents, i.e., "a\u0301" = á does not convert into {\'{a}} as it should. Interestingly enough it seems like the Java compiler would also convert "\u0301a" to á, but "a\u0301e" = áe, so clearly the combining accent should be applied to the preceding character.

Double combining accents are not supported in either HTML nor Unicode to LaTeX conversion. While some of the codes are there, the behavior is not correct. For example, b͞c or "b\u035Cc", meaning b͞c, should be converted to something clever, probably not {\textdoublemacron{b}}c which is the current result from HTML.