“There’s” or “There are”?

I wanted to get the usage of There’s clarified. I have read sentences like:

There’s a lot of projects on that topic.

It appears to me that There’s applies to a lot of projects, rather than to the projects themselves. Is that right? If that’s the case, is

There’s many students standing there.

not grammatical?

Hang on, slow down here!

There are two different matters being questioned here.

1) Why is "a lot" preceded by is

A lot in this sense describes a certain group of something, which is singular. There are many other similar words, examples:

There is a lot of people here.
There is a team of people working on this project.
There is a group of bystanders watching the accident.

2) Why is "there are" sometimes shortened to "there's"

Yes, this actually does happen. I'm not claiming it's grammatical, but people do it. Simply for the reason that you cannot shorten "there are" to "there're." In that case, you can say "there's" when meaning "there are", but this can be used only in spoken or informal English. Examples:

There are many apples on the table. => There's many apples on the table.

For further reference see Is “there're” (similar to “there's”) a correct contraction?, where the currently accepted answer states:

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.

Possibly Related:

• “There are so many” vs. “There is so many”
• There is/are one or several apple/~s?
• “Is there” versus “Are there”
• “There is/are more than one”. What's the difference?
• Should I say “there is a handful of…” or “there are a handful of…”?
• Is “there're” (similar to “there's”) a correct contraction?
• Which is correct: “There are not any employees” vs. “there is not any employee”
• How to choose verb after “There” in beginning of sentence?
• Consistency of “There is the same number of elements in… as there are in…”
• “A lot of ideas” is or are?
• Why use “is” in this sentence: “Here is your umbrella and your coat”

Neither of your sentences sounds immaculately grammatical to me, especially the second one. You’re right that people do say these things, sometimes. Here’s why.

First of all, a lot of X takes the number of X, because it can be used on both count nouns and mass nouns:

• A lot of our farmers are suffering from the drought.
  There are a lot of farmers suffering from the drought.
  

• A lot of the trouble comes from the lack of rain.
  There’s a lot of trouble from the lack of rain.

Sometimes you can start of your a lot of thinking it singular or plural before you finish it off, so that will stick in your head and you’ll leave that as the number when you get to the verb without subjecting to careful analysis. This is speech, remember, so people don’t have time to think too much. Text messages and quickly jotted Post-it® notes still count as speech, not written languages.

Next, there are legitimate situations where there is introduces a plural and nobody thinks a thing about it:

Oh look, there’s your mom and dad!

Just as they might say

Hey, here comes your mom and dad!

Even though technically, you would use come there in the plural. But here comes is too much of a set phrase, just as there is has become, to always think too much about formal agreement. Furthermore, many speakers will sometimes use there is as a generic existential that doesn’t take number into account.

How many for dinner? Why, there’s just the two of us.
There’s just three things you need to know about women, son.

This may occur in rapid, casual speech in ways that more careful writers may rewrite into the plural as needed when setting things down in formal writing.

How many for dinner? Why, there are just the two of us.
There are just three things you need to know about women, son.

Or not.