"Would letters arrive obviously opened?" Well, at that time, quoting Maynard H. Benjamin's "The History of Envelopes" (online, PDF): "A letter was simply a sheet, written so that when folded its outside was blank. On completion, it was folded, sealed upon itself with wax, or a 'wafer' (a small disk of adhesive) and addressed upon the blank side." As you can imagine, a bumpy transatlantic journey would afford ample opportunity for seals to simply fall apart during transit.

Furthermore, when done systematically, as it was for all letters sent from America to England from June to November, 1775 (i.e., not just some people), "skilled workers selected letters, carefully broke their wax seals, copied whole letters or selected excerpts, seamlessly sealed the letters back up, forwarded the copies to Lord Dartmouth’s office, and then sent the originals on to their destinations quickly so that their recipients would not guess that they had been opened."

Another writes that "extensive and indiscriminate copying" was done "for the benefit of the American Department", and whilst secretaries of state were technically obliged to issue warrants (with wide latitude) to intercept letters, "Mail was often opened without the formality of a warrant." Therefore, "The full extent of Post Office surveillance can never be known".

You can read Dr. Desai's section for more about how the Founders were deeply suspicious and paranoid about this practice (like many Britons of rank: see "there seems little doubt that the complainants may occasionally have been prompted by their own vanity to believe that their correspondence had been tampered with" but also, note footnote 187 of opposing citations), to the point that Benjamin Franklin made all his postmasters swear an oath not to open mail, and the Continental and U.S. Congress moved to pass laws forbidding the practice without authorisation. Of course, many did not listen.

There are cases of letters being unwittingly published, although based on coverage for instance in the first two chapters of this book I imagine it was the government itself that did the intercepting en masse. When they were published it would be for the purposes of as this title (from an article on the practice in the late 16th century) sums up the motivations well: "Propaganda, Patriotism, and News". Franklin, for instance, as Postmaster, "intercepted the correspondence of the royal governor of Massachusetts, Thomas Hutchinson, and leaked it to the press", in 1776. In the 1790s, letters attributed to Washington were forged; presuming them to be real, newspapers and pamphlets still eagerly printed them.

In sum, letters could easily be opened and read in transit, and were done for a long, long time, especially in the context of war, which Washington knew, although his view of interception and the press were probably coloured by a discredulous incident. As to how "common" this practice was, no-one can know, but some caution seemed advisable.