Sentences 1 and 2 are natural (except that in a first mention it's usually "I met AN old friend"), while Sentence 3 is not quite so natural (see below). Sentence 4 is correct if "broke the vase" is 'old" information. Sentence 5 is correct, if "broke the vase" is the "new" information. Sentence 6 is also correct, if "accidentally" is the "new," or more important, information. The choice has to do with the order of the "old" and the "new" information.
"By accident" occurs after the verb phrase in any case*:
--City employee used [city purchasing card] for haircut by accident (headline; see also below)
--Spot says "don't learn about electrical safety by accident"
"Accidentally," in contrast, comes before the verb phrase if the verb phrase contains "new" information.
--Lemley told detectives she accidentally "struck the delete key on her computer while (her friend's) files were on the screen,"...
--They're vacationing in Scotland and Claire is out innocently gathering the heather on the hill ... when she accidentally stumbles upon a ring of stones that bears an eerie resemblance to Stonehenge
--A city of Topeka employee accidentally used her city purchasing card in June 2002 to get a haircut at a spa in Florida (portion of a news article, see also above)
"Accidentally" can be used after the verb phrase if it is the "new," most important, information:
--So I, [an] ordinary, non-hostile citizen, have "smuggled" a knife onto an airplane at least four times since the supposed increase in security- one time during the highest threat level. And I did it accidentally because I forgot to take it off of my key ring.
--But FDA Deputy Associate Commissioner [X] nonetheless turned the names over to a group associated with the trial lawyers. He says he did so accidentally, not having noticed that they were marked as confidential.
*All example sentences are from Google.