According to Merriam-Webster it can be used as a noun to name something that is ‘below’, but I think here in your sample sentences it’s being used as an adjective, to modify the nouns letter and solution. Generally I think more correct usage would be to place it after the noun, as an adverb, e.g. ‘the information below’ or after a verb with the noun unstated but understood (see below). As an adverb it’s not uncommon, I used to use the phrases ‘see above’ or ‘see below’ in emails and reports at work with a formal tone, in brackets usually, when referring to some information or a graph or whatever.

I agree with Tom. The grammatically correct version would be 'the information below...' But, as you said, you've noticed 'below...' used in emails because it's being used in a short-hand way.