And to your question about algebra filter /filter/algebra/algebra2tex.pl needs also correct permissions to execute and because it is a sub filter of tex filter it uses executable file of mimetex from folder filter/tex and you may also need a functional perl parser - see for example http://tracker.moodle.org/browse/MDL-9252
If checking and setting of permissions and checking of paths or even symbolic links don't work then some setting of php (extra security etc restriction) might prevent creation of new images.
Check also php info from administration menu > Server
Are there any disabled functions that might prevent executing of commands ( disable_functions ... )?