Matilda, you would have got a much quicker response to your suggestion if you hadn't thrown in the phrase "(and rime while I'm at it)". I've no doubt that riming should be classed as rare - and rimed too. But rime is not so clear cut. Firstly because rime in the frost sense is used more often as a noun than as a verb. And rimed is not used very often either, whether as a verb or an adjective. But also because, as people have pointed out, rime is an older spelling of rhyme, and is fairly well known because of "The Rime of the Ancient Mariner".
For some reason, Coleridge's original spelling of rime is still being used, but not the other apparently archaic spelling in his original title: "The Rime of the Ancyent Marinere". There are a few archaic words we treat as common because people are very likely to be familiar with them. Examples are thee and thine, which are regularly encountered in old books and historical fiction and movies. But I wouldn't put too much weight on the effect of a single poem.
In recent publications, most of the uses of the word rime were as the name of a video game - for all I know this may be familiar to more people than Coleridge's poem. But in any case, I think it's safe to say that there would be a number of people not familiar with either. In texts from a few years ago in the Corpus of Contemporary American English, a lot of the examples of rime are actually OCR mis-readings of time: "If something lasts a good long rime ..." This is a sure sign that the word being searched for is not used often.
So rime, riming and rimed will all be treated as rare in future.
