From a US perspective, I think the most important thing is politeness and clarity. "Official" contractions should be fine, as long as the writing is otherwise grammatical. ("You're" is fine, while "ur" should be avoided like the plague.) (There is a slight possibility that this differs at some very formal British universities(???) but it should be fine at any U.S. university.)
I would steer clear of jokes by email, since you haven't met them before and the communication is across a language (and culture?) barrier.
When I email people I don't know who work in staff positions, I keep in mind that they're busy and often deal with demanding people or people who are upset. I usually try to keep the "polite" parts of the email at the beginning and end, so the other parts are clear in the middle.
As a prospective graduate student [from country or university, if relevant to question], I have a few questions about enrolling in the English department that I hope you may be able to help me with. If there's a better person to contact about these questions, I'd really appreciate it if you could direct me to them. [If it's a question about funding or visas, e.g., it's quite possible they would probably have to refer you to another office.]
[Possible additional clarification about what you have already tried, e.g. "I saw on the department web page that all applicants must have taken an introductory British literature course (link to that webpage), but it did not specify whether courses taught in another language ('English Romantic Poetry in Translation') or about specific areas (like 'Irish Literature' and 'Indian Literature: Colonial Era' would qualify)."]
[If you're attaching any files, have a really good reason, and explain them here. E.g. "In case it helps answer the question, I am attaching the descriptions of the British literature courses I took, with my own translation of the descriptions from [my language]. The original versions are online at [link], [link], and [link]."]
Thank you for your time. Please let me know if you need any additional information or clarification.
Thank you very much,