Suggested answer:
(1) It's is a combination of the pronoun it and the third person present tense verb form of either be or have or it + the third person past tense form of be. Its is a determiner/possessive pronoun/determiner (the genitive form of it) expressing ownership or something that belongs to something. Examples from the corpus:
It's quite strange... (it's = "it is")
I'm lucky it's worked out quite differently for me. (it's = "it has")
Its destructive effects can be traced...

In some cases it is hard to tell whether the "'s" represents is or was, e.g.
It's all a bit confusing from both sides.

(2) In the corpus 's only has the meaning of "has" in the case of he's, e.g.
He's got a nice voice and he can play anything by ear.

The 's in here's typically has the meaning of "is", e.g.
Here's your pass.

(3)Their is used as a possessive determiner, e.g.
Their hands became freed from locomotion duties...

They're is a combination of the pronoun they and the third person plural present tense (or past tense) of be, e.g.
They're good-natured people.

There can function both as a locative adverb (a) and as an existential subject (b):
(a)...we saw her mum there.
(b)There are a number of arguments...

There are 15 occurrences of there're in the corpus; it is the existential use of there in combination with a plural form of the verb be, e.g.:
There're no secrets here.

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Terminology: determiner, pronoun, possessive pronoun, possessive determiner, genitive, existential there