There are three big questions there, with the second, on profitability of the colonies, a tricky one which can plunge us into discussion and the last, on why states bother, quite broad. Others may like to weigh in on them and I'll focus on your first part, "I have been told by many acquaintances that the loss of the thirteen colonies actually BENEFITED the UK...Is there any truth in this statement?"
The historical consensus seems to be a resounding yes to the broader question of general benefit in the long run.
The most recent restating of this I found is by Maya Jasanoff in her 2012 book Liberty's Exiles. Though it is primarily on the loyalists, she sums up some of the benefits of the loss of the conflict to Britain in her opening section on what she calls the "spirit of 1783," some of which speak directly to your interest in the economic aspects in particular:
- As a result the British empire significantly expanded - with the loyalist exiles she focuses on as agents and advocates [1:12]
- 1783 serves as a dividing line between old and new empire with new focus on Asia etc., the “swing to the east” [1:12], also similar point in 2:76, which says the loss “helped to consolidate a new focus of empire, and an empire that was markedly different from that which had gone before”
- there was a renewed commitment to authority but one that “joined liberal principles with hierarchical rule” [1:13] - others have described how, in particular this is seen in the case of its flexible response to a crisis in Ireland that enabled it to remain in power there, as well as new approaches to its other dependencies
The chapter on the impact on Great Britain in The Impact of the American Revolution Abroad concludes with:
The expansionist forces released by the Revolution worked,
surprisingly, for the benefit of Britain. The growth of the American
economy made for the vastly increased profit of Britain and enabled
her to sustain a long and expensive war against France and Napoleon.
You might want to have a look over this short summary on About.com about this as well:
About.com: The Effects of the American Revolutionary War on Britain
Note: The short-term calculations of economic benefit would be a trickier thing to work out, especially because fundamental transformations are in the works that are not directly tied to the course of events in the US and would be difficult to filter out. This comes through in the opening chapter "England 1783–1846: a preview" in A Mad, Bad, and Dangerous People?: England, 1783-1846 which lays out the foundations for the "Great Transformation" that is about to take place. At any rate, the historians in various sources I scanned over not cited here all focus on the broader long-term picture of renewed trade with the US and a shift in empire, which I think is appropriate when looking at the impact of a world-historical event like this.
Citations above in form [Source:Page Number]
Maya Jasanoff, Liberty’s Exiles: American Loyalists in the Revolutionary World (New York: Vintage Books, 2012). On Google Books
Library of Congress, The Impact of the American Revolution Abroad (The Minerva Group, Inc., 2002). On Google Books
Boyd Hilton, A Mad, Bad, and Dangerous People?: England, 1783-1846 (Oxford: Oxford University Press, 2008). On Google Books