The php notices and warnings are mostly PHP and access rights configuration problems, but they don’t stand in the way of a working system. You may choose to disable the offending lines of code, it won’t break anything critical.

The real problem is that Gosmore is not found in the directory in which it is expected:

[Mon Sep 05 14:36:16 2011] [error] [client 192.168.5.102] PHP Warning:  chdir(): No such file or directory (errno 2) in /var/www/yours/api/1.0/gosmore.php on line 164
nice: ./gosmore: No such file or directory

So please investigate line 164 more closely, the variable provided to chdir() is probably not correct.