partial, and probably unsatisfactory, answer.
unicode alone can't do everything. for example, if you want an integral from
\infty, unicode has the codes, but it isn't by itself able to position sub/superscripts or limits. so at a minimum, some sort of markup and composition facility is required.
markup could as well be mathml as latex, but that's up to whoever is preparing the document.
as for whether "all" latex characters/symbols are covered by unicode, the effort made for
the benefit of stipub (see http://www.ams.org/STIX for the history of the stix project)
attempted to get as many such symbols as possible accepted into unicode. if a symbol was
requested by one of the stipub organizations, then it went onto the list, and by and large
the unicode technical committee received that request as an acceptable level of
documentation. for some edge cases (some symbols in the
stmaryrd collection or in
tipa, for example) which were not on the main stipub lists, additional documentation --
in the form
of articles or books published by recognized technical publishers -- was required, and
in its absence, no action was taken. (if someone can provide a suitable citation for
any "missing" symbol, the effort to add new symbols is ongoing.)
what did happen is that the unicode technical committee accepted the proposition that
math notation is effectively a "language", and as such, symbols in common use should be
encoded just as letters for "minor" human languages, alive or dead, are encoded. this
is what is required for mathematicians and other scientists to communicate on the web.
i am not aware that a complete list of symbols, with their visual representation and
associated unicode (and, potentially, a "tex name") exists yet. i hope that this
information can be added to the "comprehensive symbols list" (
but that is a massive undertaking (in which i am willing to participate, but haven't
yet contacted the author to that effect). and some glitches in the stix fonts, which
were the outcome of the stix project, remain to be ironed out, in particular the
location of quite a few "unicodes" in the private use area.
regarding direct use of unicodes or the "native" symbols vs. "tex names", ability to
do so depends on the engine in use. it's probably not possible with pdflatex, but
should be relatively straightforward with xelatex provided suitable fonts are available.