The approach I would take in this case, and as indicated here, is to boot into recovery mode (or "single user" mode), then fix the permissions on the /etc/sudoers file.

That answer is for Ubuntu, but the same approach should work regardless of distribution.

If you for some reason cannot boot into recovery mode, then I would suggest that you use a Live medium (CD, USB) and fix the permissions from there.

How to configure pkexec to avoid getting errors when run GUI applications?

I found two possible ways:

1. As you can see, using the following:

   pkexec env DISPLAY=$DISPLAY XAUTHORITY=$XAUTHORITY gedit
   

   will not get you any error. And this is normal because man pkexec is very clear in this matter:

          [...] pkexec will not allow you to run X11 applications
          as another user since the $DISPLAY and $XAUTHORITY environment
          variables are not set.[...]
   

   As result you can create an (permanent) alias (this is the simpliest way):

   alias pkexec='pkexec env DISPLAY=$DISPLAY XAUTHORITY=$XAUTHORITY'
   

2. Or, (again) as man pkexec says:

          [...] These two variables will be retained if the
          org.freedesktop.policykit.exec.allow_gui annotation on an action is set
          to a nonempty value; this is discouraged, though, and should only be
          used for legacy programs.[...]
   

   you can create a new policy file in /usr/share/polkit-1/actions named com.ubuntu.pkexec.gedit.policy with the following xml code inside where the most important thing is to set org.freedesktop.policykit.exec.allow_gui to a nonempty value:

   <?xml version="1.0" encoding="UTF-8"?>
   <!DOCTYPE policyconfig PUBLIC
     "-//freedesktop//DTD PolicyKit Policy Configuration 1.0//EN"
     "http://www.freedesktop.org/standards/PolicyKit/1/policyconfig.dtd">
   <policyconfig>
   
     <action id="com.ubuntu.pkexec.gedit">
       <message gettext-domain="gparted">Authentication is required to run gedit</message>
       <icon_name>gedit</icon_name>
       <defaults>
         <allow_any>auth_admin</allow_any>
         <allow_inactive>auth_admin</allow_inactive>
         <allow_active>auth_admin</allow_active>
       </defaults>
       <annotate key="org.freedesktop.policykit.exec.path">/usr/bin/gedit</annotate>
       <annotate key="org.freedesktop.policykit.exec.allow_gui">true</annotate>
     </action>
   
   </policyconfig>
   

How to tell it to not ask for a password after the first time applying it to a command?

For these three setting tags: allow_any, allow_inactive and allow_active from the policy file, the following options are available:

• no: The user is not authorized to carry out the action. There is therefore no need for authentication.
• yes: The user is authorized to carry out the action without any authentication.
• auth_self: Authentication is required but the user need not be an administrative user.
• auth_admin: Authentication as an administrative user is require.
• auth_self_keep: The same as auth_self but, like sudo, the authorization lasts a few minutes.
• auth_admin_keep: The same as auth_admin but, like sudo, the authorization lasts a few minutes.

     ^{Source: Polkit - Structure - Actions}

So, if you use auth_admin_keep option (or, as applicable, auth_self_keep), pkexec will not ask for a password again for some time (by default this time is set to 5 minutes as I checked). The disadvantage here is that this thing is applicable only for one - the same - command / application and valid for all users (unless if it is overruled in later configuration).

Where to save the configuration file if not yet existing?

Configuration files or polkit definitions can be divided into two kinds:

• Actions are defined in XML .policy files located in /usr/share/polkit-1/actions. Each action has a set of default permissions attached to it (e.g. you need to identify as an administrator to use the GParted action). The defaults can be overruled but editing the actions files is NOT the correct way. The name of this policy file should have this format:

  com.ubuntu.pkexec.app_name.policy

• Authorization rules are defined in JavaScript .rules files. They are found in two places: 3rd party packages can use /usr/share/polkit-1/rules.d (though few if any do) and /etc/polkit-1/rules.d is for local configuration. The .rules files designate a subset of users, refer to one (or more) of the actions specified in the actions files and determine with what restrictions these actions can be taken by that/those user(s). As an example, a rules file could overrule the default requirement for all users to authenticate as an admin when using GParted, determining that some specific user doesn't need to. Or isn't allowed to use GParted at all.

     ^{Source: Polkit - Structure}

Is there a GUI application to configure pkexec usage?

From what I know, until now (18.01.2014) doesn't exist something like this. If in the future I will find something, I will not forget to update this answer too.

Your error is being generated by pkexec itself as your call syntax is incorrect.

You used os.cwd() to get the CWD but then you prepend it with a . which invalidates the full path.

Effectively the equivalent of this:

amos:/tmp$ ll simple.sh
-rwxr-xr-x 1 amos amos 73 Mar 16 11:51 simple.sh*
amos:/tmp$ pkexec .$PWD/simple.sh
Cannot run program ./tmp/simple.sh: No such file or directory

Really there is some kind of hack to use pkexec instead of sudo to launch application without specific PolicyKit .policy file. We need to specify missed environment variables as shown below:

pkexec env DISPLAY=$DISPLAY XAUTHORITY=$XAUTHORITY kubuntu-driver-manager

Experimentally I discovered that it needs only one /usr/share/polkit-1/actions/org.freedesktop.policykit.policy file for operation.