I am thinking about the history of the verb "have". Why is the verb "have" used as an auxiliary verb in the perfect tenses? When did it start to be used that way?
According to the OED (have, sense VI):
The have-perfect in English apparently arose as a reanalysis of uses such as I have my work done ‘I have my work in a done or finished condition’; the complement done was reinterpreted as part of the verb phrase, a process which was reinforced by a lack of fixed word order and the possible transposition of object and participle, i.e. I have done my work. This development appears to have largely taken place before the written record. Even in early Old English, in the majority of examples with transitive verbs the past participle is not inflected to agree with the object. Despite occasional ambiguity, there are few Old English examples in which the past participle must be regarded as a complement rather than as part of a perfect construction.
In Old English, the have-perfect is not only established with transitive verbs, but also with intransitive verbs expressing action or occurrence, while the perfect of intransitive verbs expressing change of state or position is usually formed with be. From Middle English onwards the perfect with have gradually becomes more common in these verbs, and is the predominant form by the early 19th cent., except in contexts where the focus is on resultant state (for example, she is gone is still typically used to express state, while she has gone expresses action; such usage is now, however, quite limited). In early Middle English the have-perfect also extends to verbs denoting ongoing states or conditions, and to the verb to be.
This "have-perfective" is one of the most significant features of the Standard Average European sprachbund—in other words, it's a feature that's shared by a lot of not-necessarily-related languages in a particular area.
There's some disagreement about where in the sprachbund it originated, but personally, I like the theory that it arose in Vulgar Latin. Like the OED described, it seems to involve re-analysis of sentences like (litterās scriptās) habeō "I have (written letters)" as litterās (scriptās habeō) "I (have written) letters". The ancestor of English "have" (and German haben and such) was then an easy equivalent for the etymologically-unrelated Latin habēre.
Latin was written much earlier than Old English, and in the Latin record we can see this construction evolving, from marginal uses in Classical texts (there's one attestation in Vitruvius that scholars disagree about how to analyze) to a variety of Vulgar attestations to becoming the standard in Romance. In Classical Latin, a past aoristic verb ("I wrote") and a present perfect verb ("I have written") look identical, so there was plenty of reason for Vulgar Latin to develop a way to distinguish the two, leading to the reanalysis mentioned above.
It's also possible that this development started in Germanic or somewhere else and spread from there to Latin. Since we don't have early written records of these other languages, this is a difficult claim to falsify—as the OED mentions, the oldest written forms of English show this construction. But, the earliest documentation of it seems to be in Vulgar Latin, and I find it very easy to believe it spread from Romance into Germanic.