What's the difference between "recently" and "lately"?

Why is it "there's" not "there're"? Because "a thousand reasons" looks like plural to me and "there're" seems to make sense here.

You are correct, "a thousand reasons" is plural, so "they're" should have been used here instead of "there's" for this sentence to be grammatically correct.

What does "all" mean?

'All' in this context, means 'the one thing', and it's used to emphasize the task that they're supposed to do, catch the train. Ex: All I want for Christmas is you. Ex: All I ask is for some respect.

What does "here we go" mean in this situation? and why did he say "we" not "I"?

He says "here we go" because "here we go" is an idiom that is said when someone is going to do something they are nervous about. Therefore, "I" should not replace "we" here because he is using an idiom. Ex: I'm about to have a job interview. Here we go!

Since it's several schools she's referring to, isn't she supposed to say "there are" instead of "there is"?

Absolutely! This is one of the most commonly-made grammatical mistakes by native speakers. The correct way of saying this is "there are" or "there're," but many people make the mistake of saying "there is" or "there's." You should also make note to avoid making this mistake when speaking. Ex: There's a bear in front of our car. (There is + single form noun) Ex: There are bears in front of our car. (There are + plural form noun)

What does "as a plan B" mean? Can I use other initials instead of B?

"Plan B" is used to mean an alternative plan of action for use if the original plan fails. You could use any letter but generally "b" is used because you're referring to a second plan. "Plan A" refers to the original plan and any other letter would have the nuance that it is the same number plan as that letter in the alphabet. Ex: A: Well, there's always Plan B. And if that fails, Plan C and D. B. How many plans do you have? A. I'm covered all the way up to G. Ex: Let's just go with Plan B.