Beg and implore refer to asking someone for something (aid, mercy, pardon) with urgency or in desperation. The essential element of beg is asking for something one is not owed by the one asked, from a position of powerlessness. We say, “Don’t make me beg,” but not “Don’t make me implore.” With implore, the focus is on the emotion and manner of expression with which someone begs. A person may implore with their eyes. Implore suggests begging with great urgency, in an affecting, piteous way. (The Latin word it derives from means "to plead tearfully.") Beg can convey a wide range of intensity–it can have the mere formality of “I beg your pardon,” the humility and earnestness of someone begging for alms, or the fervent emotion of implore.
Prestige is status or influence a person or institution possesses in the eyes of others by virtue of success, wealth, or position (professional prestige, international prestige). The sources and qualities of cachet can be less tangible–hence, perhaps, the common phrase “has a certain cachet,” where “certain” means “definite but not specified.” Cachet is found most often in the context of popular culture, consumer and marketing culture, art, and fashion. Associations, especially with celebrity names, lend things cachet, and the associations often attract and impress a particular demographic or other group (a label carrying cachet among Spanish wine lovers, a director whose use of indy music in his films has earned him cachet).
Select refers to carefully choosing one or more things or people from a group, according to preference or some criteria of superiority (select one of the candidates, select a necktie to wear to the event). Cull suggests selection or gathering together of a number of things or people, typically by combing through a large quantity. It is often used in the context of research or curation–he culled the footage from video archives. When using cull, however, keep in mind that cull is one of those odd words, like cleave, that has two nearly opposite meanings: culling also means selection of things you want to eliminate (cull out the duplicates in the database, cull the deer population).