OK for starters you're getting errors relating to the error log not being writeable so you need to sort that out. Go to
Admin Panel --> PHP Error Log and set an actual folder then make sure that folder is writeable.
As for alternatives for editing php.ini you can try using ini_set in
include/functions.php just add this line near the top of the file:
PHP Code:
ini_set("date.timezone", "YourTimezoneHere");
Replace
YourTimezoneHere with an appropriate timezone from
here.