Are the free mySQL GUI tools adequate for administering a mySQL db running 'vendor' supplied applications?
Are there other tools you would recommend that make your life easier?
There's:
There's also MySQL Administrator which is provided by mysql although I've never used it.
Other tools: For backup:
i'm surprised maatkit was not mentioned yet. it does not provide any fancy gui applications, instead it gives dba set of command line tools that can be used to analyze mysql logs, diff and synchronize content of databases, run faster database dumps [ by executing parallel queries ]. you can read more here.
another useful tool - xtrabackup - free & open source hot backup of innodb databases.
Navicat for MySQL (there are also Oracle and PostgreSQL versions too). Not free though. US$179 for the downloadable version ($189 for the CD). Highly recommended for all MySQL DBAs and developers using MySQL.
There is a trial version available as well.