Our host seems to think the myriad issues we've been having may have to do with some of the descriptions being "too long" but we can't find any specifics on how long too long might be.
Does anyone know if there is a character limit on the description?
Yes, the product description in Magento 1.x and 2.x is limited to 64Kb due the limitations of the MySQL field type "TEXT", which is limited to 2^16 bytes, see MySQL Storage Requirements
Depending on your requirements a possible workaround could be to store the necessary information inside a CMS block, create a own product attribute and reference the CMS block by the new product attribute. But this solution has also some limitations, e.g. the information won't be indexed by the product search, etc.
The other option would be to create a new Magento data type like catalog_product_entity_text, but has huge implications on the architecture of the system.
The last and from my perspective most unsafe solution would be to change the MySQL type from text to mediumtext of the value attribute of the table catalog_product_entity_text, which is described in the answer of Product attribute as textarea .. character limit reached. This could have some negative impacts on the whole system (e.g. indexing, etc.).