“his dry nurse, or his cook”
A 'dry nurse' cared for a young child without breastfeeding — managing feeding, washing, and watching — as distinct from a 'wet nurse' who suckled an infant. Evans is piling up Mistress Quickly's domestic roles (nurse, cook, laundress, wringer) to convey that she does everything in Doctor Caius's household.
“his laundry, his washer, and his wringer”
'Laundry' here means a laundress — a woman whose job is washing linen — not a room or building. A 'wringer' is one who physically wrings (twists) wet cloth to squeeze out the water, a separate step in hand-washing before wringing-machines existed. Evans's list of job-nouns treats Mistress Quickly as a one-woman domestic staff.
“desire and require her to solicit”
'Desire and require' is a legal doublet — the pairing of near-synonyms to cover every shade of meaning, common in formal documents and ecclesiastical correspondence. Evans, as a Welsh parson, instinctively reaches for this church-legal register even when giving a servant a simple errand. The redundancy is part of his comic formality.
“pippins and cheese to come”
Pippins are a variety of apple — the name comes from the French 'pepin' (seed), applied to apples grown from seed rather than grafted. Cheese and apples eaten together was the standard Elizabethan dessert course; Evans is telling Simple he is interrupting his own after-dinner fruit-and-cheese course and must get back to it, grounding the play's opening moments in ordinary Windsor domestic life.