The short answer to your last question is no. Rhyn is not an orchid abbreviation. Rhy. is as close as you can get but it stands for Rhynchostylis. When they combine two or three genera together the abbreviation tends to come from all three, not just the first of the series. Also of note is when they combine more that three genera together, they do not form these multi-letter words like Brassolaeliocattleya but rather chose a word that ends in "ara". For example, a Brassavola X Cattleya X Laelia X Schomburgkia X Sophronitis is a Fergusonara. Another example is Potinara, which is the name of a hybrid of orchids which have Brassavola, Cattleya, Laelia and Sophronitis as parent genera.
Now, back to your request:
Rhyncholaeliocattleya [Rlc.] Ports of Paradise
Rhyncholaeliocattleya [Rlc.] Greenwich
Rhyncattleanthe [Rth.] Zul
I'll get the rest later tonight as I have to dash off to a funeral.
Jim