I can start with how it was/is done in Russia.
Logic was taught in Russian gymnasiums as a separate subject in late 19th century. When Bolsheviks came to power they pulled logic out of the curriculum. Logic was reinstated in Soviet schools in late 1940s only to be abolished again by the end of 1950s when Khruschev took over. AFAIK, there is no separate logic course in modern Russian schools, at least it is not mandated by the federal ministry of education, some specialized schools may teach it as an elective. There is a separate course of logic in universities, I don't know whether it is mandatory for everyone, or depends on a major.
Proofs are common in Russian algebra and geometry courses (both subjects are mandatory, start from 7th grade and continue until the senior 11th grade). Most commonly they are direct proofs based on the material learned earlier, so their application is obvious and does not require much of an explanation how the proof works. Traditionally, proofs in Russian schools are of "free text" variety, not a two-column system used in many U.S. schools.
Terms direct, inverse, converse and contrapositive are introduced in 7th grade geometry in relation to theorems, not to abstract logical propositions. There is a simple explanation of why contrapositive theorem is equivalent to the original one, similar to the one shown in Wikipedia. This fact is harnessed when using proof by contradiction.
Older textbooks may have a separate, albeit tiny, section dedicated to proofs, like Kiselev's Planimetry, which was used in Tsarist Russia, then was updated in 1938 and was used until 1956, when it was replaced with "New Math" textbooks. These new textbooks proved disastrous, so many teachers continued to covertly use Kiselev's textbook. The textbook has been re-issued in 2004 and again in 2013. Here are relevant pages from the English translation, which includes problems; in Russian edition problems comprise a separate problem book.
Other methods that feel more or less self-evident, are sometimes used: proof by construction, proof by exhaustion, Dirichlet's box principle.
9th grade algebra course introduces proof by mathematical induction, although in some textbooks it is marked as optional.