While technically wrong, it’s common in colloquial English for even educated speakers to use “there’s” with a plural noun. In this case, we normally use the contraction “there’s” rather than the full form “there is.” I suppose this is a bit like the grammar of the French “il y a”. Also, it should be noted that if we list more than one type of noun, the first noun determines the number, for example, it’s correct to say “There is a school and three restaurants.” It would be weird to use “are” when the following noun is singular.

In the case where you use the 'a lot' construction, you could argue that 'a lot' becomes a singular collective noun. As the others have said, it is very common to say it that way. By 'collective noun' I mean that 'a lot' functions the same as 'a box' - a single (conceptual) object that contains plural objects. There is a box of chocolates under the bed. There is a lot of balls under the bed. This is similar to other collective nouns, e.g. 'my family'. Although a family is a singular noun, we can use both singular, or plural to refer to our family, usually depending on whether we are referring to the group, or the members individually.

You are correct. It should always be there are when followed by plural nouns, there is no exception i know of. However, in informal speech it will generally go unnoticed (the speaker may not even be aware). I am often probably guilty of this myself!

When you add the apostrophe, followed by the letter "s," all you've done is shortened it from 2 words into 1. There's = There is (sounds different, but means the same) Its the contraction of the word, meaning it now must have an apostrophe. Using this in every day talk is fine, and for elementary level writing reports. However, some will argue for higher academic reports, it is best to use both words instead of the contraction, "there's." I hope this helps.