In The Company They Keep, Diana Pavlac Glyer demonstrates (against claims by Humphrey Carpenter and others to the contrary), that there are clear signs in their individual works of mutual influence. Glyer is continuing to investigate the creative power of collaboration, among the Inklings and also in other literary circles, as a source of inspiration and guidance for writers of all kinds.
Finally, as your last question suggests, there is a long history of informal intellectual clubs. We mention the 18th-century Scriblerus Club, which included Alexander Pope, Jonathan Swift, and John Arbuthnot as well as Samuel Johnson's dinner-and-discussion circle, called "The Club," and we compare the Inklings to the older, convention-busting Bloomsbury Group. This is just the tip of the iceberg, of course. Think of Wordsworth and Coleridge; their Lyrical Ballads may be regarded as inaugurating the literary and spiritual movement to which the Inklings, as latter-day Romantics, were heir. And though "clubbability" is a British phenomenon, with its heyday during the expansion of the British empire, there are plenty of examples to consider on our own shores — the Metaphysical Club, for example, to which Louis Menand devoted a wonderful book.