Here is the thing about validating : it’s a great goal to have but sometimes you have to complete a balancing act between what you need to achieve in terms of design, time available, browser compatibility with the strict validation.
Of course most of the times you should avoid hacks (example : the
underscore hack used above) to ensure all browser render OK. For instance the moz properties are proprietary thus not standard compliant, the proper opacity property is not understood properly by IE for instance and this forces you to use hacks or improper syntax if you want rendering to be consistent... there are other examples.
I am not sure full validation was a *strict* priority when building MODxHost template. Nor should it be, in my opinion. Standards are base guidelines, which of course you should follow but I think it should never supersede improving the user experience or making sure rendering is consistent. In a way, as we say in french, stick to the spirit, not the letter of the law
But others will have different opinion

Maybe I am just too lazy to take the time to make all things fit, or I am not paid enough to take the time to do it

I’d be curious to know what my fellow MODxer will say about that...
Anyway, let me help the good cause by adding a resource :
This page is a treasure of errors or reason why validation can fail (The underscore hack is explained in 10.) If you can build a 100% validating variant of the MODxHost template, well, that’s great !
But remember MODx is not responsible for non-validating templates : you (or the people who build the template) are. MODx is great for designers because it doesn’t strap you into a design pattern, but also because it doesn’t output some badly hardcoded HTML into your templates like some other CMS do (I won’t throw names here

).
As for the [tt]<quickedit:content />[/tt] issue, it can be fixed you don’t *have* to use those tags just prepend your TVs with # (write [tt][*#content*][/tt] in your template instead)
As for AjaxSearch : I have no answer right away.