You need a later LR version to have the RAW from your cameras supported: https://helpx.adobe.com/camera-raw/kb/camera-raw-plug-supported-cameras.html
LR is available as subscription or as standalone perpetual licensed Software.
LR can handle JPG and RAW just fine. There are options to tell it how to handle the two if both are available for the very same photo.
Am I correct that the issue is that a folder with both JPEG and RAW causes Lightroom to not be able to import anything?
No, that is not correct. In all versions of Lightroom, there is an option you can set in the Preferences that treats JPG as sidecar files to the RAW, one setting allows both JPG and RAW to be imported, the other setting allows only the RAW to be imported.
Beyond that,is it true that any Lightroom product that would be able to handle both is only available to me on a subscription? Thanks again!
No, you can buy either subscription versions of Lightroom or standalone versions of Lightroom, which can both allow you to import.
The 5D is supported from Lr 1.0 onwards; the 1D-X from 4.1 onwards.
Adobe doesn't sell old versions of Lr though, so unless you can find a copy on Fleabay or the like, you'll need to upgrade to Lr 6/CC 2015.
Or - convert your files to DNG using the free Adobe DNG Converter, which Lr 2.7 will be able to convert.
But it's a waste of the 1D-X not to feed its files into the latest Lr - image quality from later versions of Lightroom is much, much better than you'll ever get from 2.7,
I agree with Keith. The algorithms for highlights, shadows, whites & blacks are far superior in the current version when compared with recovery & fill-light for example. You can upgrade to LR6 from any previous version.