Rewriting Controller vs. Adding New Controller to Existing Module in Magento 1

This may depend on how the extension is overriding the controller but here goes. You need to add in your module's config.xml file, this as a child of the <config> node.

<frontend>
    <routers>
        <checkout>
            <args>
                <modules>
                    <Your_ModuleNameHere before="Community_ModuleNameHere">Your_ModuleNameHere</Your_ModuleNameHere>
                </modules>
            </args>
        </checkout>
    </routers>
</frontend>

If the community module overrides the cart controller in the same way as described above then it works. If it uses the 'old method' with <from> and <to> tags then it won't work.
(and I have no idea on how to override that).

[EDIT]
Make sure that your controller is also named CartController and it's located in Namespace/Module/controllers/CartController.php. Also you might want to make your local module depend on the community module but I don't think this is mandatory. Also make your controller class extend the community module controller.

I find the whole XML thing difficult so I am sympathetic to your cause here.

It looks to me like there is a discrepancy with your class name and your block XML

ie

class Crusader_CLHomePage_Block_Qwerty extends Mage_Core_Block_Template

and

    <block type="core/template" name="homepage" template="crusader/home.phtml">
        <block type="crusaderhome/qwerty" name="homeads" as="homeads" template="crusader/homeads.phtml" />
    </block>

Let's start with the block type and work backward. I think here I would expect to see something like

<block type="crusader_home/qwerty" ...

and that type would refer Magento to the class

Crusader_Home_Block_Qwerty extends Mage_Core_Block_Template

Continuing to work backward on this tack, the file app\code\local\Crusader\CLHomePage\etc\config.xml would need to be modified to something like

<global>
    <blocks>
        <crusader_home>
            <class>Crusader_Home_Block</class>
        </crusader_home>
    </blocks>
</global>

It points Magento to all classes starting 'Crusader_Home_Block_*'. And now I think you need to rename your module Crusader_Home instead of Crusader_CLHomePage and change the folder names accordingly...

AND / BUT I strongly recommend you change your module name. Only ever use one capital letter at the start of the module name not three and don't put capital letters in the middle of the name either.

You may want to look into some of the tools available for generating modules. I find them very useful for avoiding typing errors. I use this one by the very clever people at MGT Commerce (it has simplicity on its side) and this one is all-encompassing and an astonishing piece of coding by Silk Software.

***EDIT following comments below

Additionally based on the error log I think the problem is that Magento can't find your class file. So double-check the caches are refreshed or off, particularly XML cache, and that the compiler is off or re-enabled.