Off the top of my head I would say that dashes tend to be used where the parenthetical remark is a brief interjection -- to be passed over quickly -- while parentheses are more often used in a long sentence (in order to reduce the confusion that can result if a sentence contains more than just a couple of clauses) or if the interjection is on the lengthy side (for instance, in order to explain something that is complicated or is only marginally related to the main topic of the sentence).
I would never write a sentence structured like example 2 because it lacks a 'which' or 'that' to introduce the parenthetical comment. What you have done there is in effect to embed one sentence in another sentence to which it has no grammatical relationship. Of course that is also true of examples 1 and 3, but in these cases the dashes and parentheses are long-standing conventions that allow the reader to absorb the interjection without interpreting it as an unwarranted interruption.
If you wanted to write example 2 using commas, you would have to say something like "I like dogs, two of which I owned as a child, because their company is always pleasant" or "I like dogs, of which I owned two as a child, because their company is always pleasant." (For what it's worth I think the second is stylistically preferable, though I find it hard to say why.)
If you wanted to use a semicolon, you would have to break the sentence into two: "I like dogs; I owned two as a child. This was because their company is always pleasant." The reason for this is that using a semicolon would break the grammatical link between the first clause of your example sentence ("I like dogs") and the last ("because their company is always pleasant).
Reply from Erik Kowal ( - England)