shortened version of a word, syllable or word group
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What are contractions (words)?
What is the difference between an abbreviation and a contraction?
What is the difference between a contraction and a portmanteau?
There're is common in speech, at least in certain dialects, but you'll rarely see it written. If I were being pedantic, I'd advise you to use there are in your example, because there is is definitely wrong, so there's could be considered wrong as well. But a huge number of English speakers, even those that are well-educated, use there's universally, regardless of the number of the noun in question, so you will probably not receive any odd looks for saying or writing there's, and if you do, just cite the fact that it can't be incorrect if a majority of people use it. As for me (a native New Englander), I use both, but may use there's in place of there're if I'm speaking quickly.
I don't think "there're" is ever going to fly -- it's not so much a contraction as a simple elision. The only thing being dropped is a glottal stop, which isn't a "real" sound in English.
From a strict prescriptivist grammar and usage standpoint, "there's" used with a plural is wrong. But in spoken language (which is the real language, squiggles on pages and screens are no more than an approximate rendering) we need to be careful with prescriptivist tendencies. It may offend the grammarian's ear, but the fact that a very large number of native speakers -- likely a preponderance of them -- make exactly the same "mistake" indicates that there is something else going on.
Remember that the rules of English, as we received them in school, are only an approximation of the real rules of the language, and that many of those rules were imposed in the 18th and 19th centuries by well-meaning scholars who aimed to make English a respectable, consistent and properly-documented language. It has never been such.
Is it because of the end of the words "-re"?
Also, "let's" is a contraction of "let us"?
It's an informal usage, but many native speakers have no problem at all with constructions like...
There's two ways this can go.
...even though they would balk at the full form There is two ways this can go.
By traditional rules of grammar, obviously, it's "incorrect". But in the contracted form it's perfectly normal in informal speech. I don't think anyone would be pedantic enough to suggest you should get out of the habit of using a natural form you're quite comfortable with.
Once a form is contracted, it starts getting glued together and agreement phenomena affect it less and less, until you have something frozen like ain't or wanna.
There, since it is a dummy, has no meaning, just a function -- to indicate that there's a noun phrase to pay attention to coming after the verb. There is neither singular nor plural, so the question of the auxiliary is problematic, since it has to be pronounced before the noun phrase it might have to agree with. This causes problems.
So, what happens is that, since the subject and the verb are both meaningless and predictable, there's no real reason to worry about the official number of a contracted auxiliary verb with a dummy subject. And, therefore, in speech (i.e, real language), many people are comfortable with there's in both singular and plural.
Not to say that there're doesn't occur; but it's less common, if only because it's an extra syllable, which is unwelcome in a contraction.