Do the dropdown search and replace lists work between sessions to be accessed?
Not totally sure what you mean, if you mean the dropdown lists on the entrys in the search/replace dialog get saved and restored when Geany is shutdown and restarted, then no.
Is v2.1 using a single colorschema folder for user and superuser?
There is the system colorscheme directory (usually /usr/share/geany/colorschemes/, or the same under /usr/local/ if you installed manually)
Or `$prefix/geany/colorschemes` if you built it somewhere else
Unrelated note: I advise against running Geany as root.
+100000000000000000000000000000000 Ubuntu (and descendents) broken sudo screws up which $home is used, so it depends on which distro you use what the answer to your question is, but its not different to 2.0 if thats what you mean.
Can I disable bracket highlights? Where?
See https://github.com/geany/geany/issues/3555