Magento – page/1column.phtml VS

Since you are using your own controller you don't need to only use the default handles. Based on the logged in status you could add in your infoAction method

 $this->getLayout()->getUpdate()->addHandle('mymodule_customer_info_logged_in');

or

 $this->getLayout()->getUpdate()->addHandle('mymodule_customer_info_logged_out');

and then in your layout.xml file use

<mymodule_customer_info_logged_in>

and

<mymodule_customer_info_logged_out>

-- addition after your edit below --

Seeing that you don't want to edit the controller (since it is a 3rd party extension), I would create a separate extension that only observes controller_action_layout_load_before

    $update = $observer->getEvent()->getLayout()->getUpdate();
    $handles = $update->getHandles();

    if (in_array('custommodule_customer_info', $handles)) {

        //code to add the custom handles based on login
    }

Why changing the root template doesn't work

Both

Mage_Cms_IndexController::indexAction()

and

Mage_Cms_IndexController::viewAction()

which are responsible for displaying the default homepage and a CMS page respectively call a helper:

Mage::helper('cms/page')->renderPage($this, $pageId)

If you jump into the helper (located at app/code/core/Mage/Cms/Helper/Page.php) and follow renderPage() to the protected method _renderPage() you'll see that Magento is checking two times for a root template (Magento CE 1.7.0.2):

if ($page->getRootTemplate()) {
    $handle = ($page->getCustomRootTemplate()
                && $page->getCustomRootTemplate() != 'empty'
                && $inRange) ? $page->getCustomRootTemplate() : $page->getRootTemplate();
    $action->getLayout()->helper('page/layout')->applyHandle($handle);     
}

and

if ($page->getRootTemplate()) {
    $action->getLayout()->helper('page/layout')
        ->applyTemplate($page->getRootTemplate());
}

Both calls happen after layout handles like "cms_page" and the like are processed, so you are out of luck here.

What you can do to change the root template

There is an event cms_page_render which you can use for adding your own XML layout handle on CMS pages. Create your own extension (I'll spare some details here) and configure the event observer in your config.xml:

<?xml version="1.0"?>
<config>
    <modules>
        <Emzee_Cms>
            <version>0.0.1</version>
        </Emzee_Cms>
    </modules>

    <global>
        <events>
            <cms_page_render>
                <observers>
                    <emzee_cms_page_render>
                        <class>emzee_cms/observer</class>
                        <method>cms_page_render</method>
                    </emzee_cms_page_render>
                </observers>
            </cms_page_render>
        </events>
        <models>
            <emzee_cms>
                <class>Emzee_Cms_Model</class>
            </emzee_cms>
        </models>
    </global>
</config>

Add your event observer:

<?php

class Emzee_Cms_Model_Observer
{
    public function cms_page_render(Varien_Event_Observer $observer)
    {
        $action = $observer->getEvent()->getControllerAction();

        $actionName = strtolower($action->getFullActionName());
        $action->getLayout()->getUpdate()
            ->addHandle($actionName . '_after');
        return $this;
    }
}

Finally, add your new layout XML handle (e.g. in your local.xml):

<?xml version="1.0"?>
<layout version="0.1.0">
    <cms_index_index_after>
        <reference name="root">
            <action method="setTemplate"><template>page/1column.phtml</template></action>
        </reference>
    </cms_index_index_after>
</layout>

You can use this method as well to add a cms_page_view_after handle or create page specific handles as cms_page_render passes the $page object to your observer.

Why you can't add a block to the 'reference left'

Are you sure that the template you're using has a left column? This question may sound silly but the default "2 columns with right bar" layout for example only offers a 'content' and a 'right' area. I can add blocks to the right column using cms_page without problems so this could be the problem.

In general, you can only easily add blocks to references and echo them if

• the chosen root template uses the block you're referencing (see app/design/frontend/base/default/template/page/*.phtml) and
• the block you're referencing either is of type core/text_list, calls $this->getChildhtml() without arguments or does something else to echo all child blocks.

Without further details, I can't tell you why your blocks are not echo'd in the left or right column.