From a prescriptive grammar standpoint, you're correct. From a descriptive standpoint, I'm not sure how common that contraction is, but I've heard it before and offhand it seems like it gets used in some regions.
Phonologically, it makes sense that it would gain traction as it's a means of avoiding the effort of the 'ere are' vowel combination. It's an addition rather than an elision, but the underlying motivation of saving effort is the same.
As a non-native English speaker myself, I still find native using contraction on "there's two cats" counter intuitive. Sometimes I use the correct grammar "there are two cats" but then it sounds too formal for the native to heard it. Then I have to adjust with the "wrong" way of saying it.
There’s nothing grammatically incorrect about “there’s”. The oversight is in confusing a contraction with an abbreviation. “There’s” is a contraction for either “there is” or “there are” and the precise one is given by context. It is not a mere abbreviation for “there is”.
Written contractions are meant to faithfully represent spoken English, in which people indeed say “there’s” for both the singular and the plural.
"Have", obviously. That's the only one that goes with "do", being in the present tense. Otherwise it would be "I did (get them)".
Not that it matters anyway, since "have got" is a weird double-barrelled construct: It means exactly the same as just "have" or "got" on their own, so take your pick.
To be fair, I think "do you have them?" would be more common for a lot of English speakers ("have you got" sounds British to me as an American, but it's possible that this is just a regional American thing). I'm not sure I would either think fast enough to care enough to tailor my automated response to a question like that based on the exact phrasing of the query.
I think out loud you'd be more likely to hear "here're your donuts" rather than "here's your donuts"), but when written, here're looks way worse. Language (written, spoken, and otherwise) is interesting and resistant to fitting into nice, neat, tidy boxes.
