What is the best way to make an opening to a Russian language motivation letter to be used in applying for a university scholarship? is "уважаемые господа и дамы" good enough when addressing several unknown people or a whole department formally?
"Уважаемые дамы и господа" (esp. don't do "Господа" as it's not gender-inclusive and you don't know if you may offend someone who reads it or not) sounds way too pompous and unmodern in Russian! Almost associates with "Товарищи", which is funny but not really a good thing. We don't have such a naturally and widely accepted opening as is "Dir Sir, Madam" in English. This phrase can only be used if you are hosting an official event (like a gala) or writing an extremely official letter. With motivational letter you want to be official, polite, but kind of connect, since you are showing a less formal side of your personality and achievements (which slipped in your resume). In case you know directly the name of a person to whom you are writing (which is always a plus to know for motivational letters, but not always possible) you could say, e.g.:
Уважаемый профессор Иванов,
Anyhow, in my opinion, if I would write a letter in Russian (which is not really common though - you can hardly find any googable examples:) ), then I would start addressing by using the dative case, as if answering the sentence "to whom it may concern": "Кафедре информационных технологий", "Комиссии по рассмотрению стипендий" (I am just coming up with random names, you have to specify directly what's the name of the institution to which you are applying).
Уважаемые дамы и господа is a more natural order (ladies first). Уважаемые коллеги (Dear colleagues) is also a very acceptable, and probably even preferable option - provided your age and academic credentials (even if relatively modest) justify this. If you have a degree, you can address them as colleagues. For an undergraduate, however, it would mean shooting way above his/her station.