Learn English – the difference between ‘I’m going to go to work’ and ‘I’m going to work’

Your first example is idiomatic; your second is not.

"Late getting back" is an elision of "late in getting back," where the gerund is used as complement to the preposition. Here, "in" carries a meaning something like "related to."

I was late in realizing that he was secretly a werewolf.

"To" does not have this meaning. The preposition "to" is used with "late" only where the complement is a noun:

I was late to the party.

I was late to my own wedding.

or with the adverb "too" to indicate that the lateness was excessive:

I was too late to save her.

He arrived too late to meet the famous molecatcher.

"So" does not work in these sentences the way that "too" does, so your second example would not be idiomatic in most English dialects.

I looked up put in the NOAD dictionary, and found this definition:

move to or place in a particular position

Based on that definition, I can understand why you think these two words might be synonymous, but they are not!

The verb put is used in a local context (that is, rarely much further than a room or house), whereas send is used in more of a long-distance context.

For example, for put, we would say:

Put the sofa over in that corner.
Put your dirty plate in the sink.
Put your trash in the garbage can.
Put that laundry basket back downstairs next to the dryer.

whereas for send, we might say:

I am going to send these cookies to Aunt Erma out in Arizona.
Please send me your application via email.
Let's send a ship to the Mediterranean Sea.
We need to send more medical supplies to Uganda.

With mail, mail is almost always sent, unless we are are talking about the individual letters, which can be put anywhere after they have arrived:

I just picked up the mail; where do you want me to put it?
Oh, please put the mail on the desk over there. Thank you.

Of course, English being English, it's not always that straightforward. In the 1960's, was the goal to
put a man on the moon, or to send a man to the moon? According to my guidance, send would be
the better word. So why did John F. Kennedy say in a speech:

I am delighted that this university is playing a part in putting a man on the moon.

I think that's because send sometimes has a connotation of being a one-way trip. We don't expect Aunt Emma will return those cookies, or that Uganda will send back the medical supplies. That's what president Kennedy was emphasizing in another speech, when he said:

...of landing a man on the moon and returning him safely to the Earth.

The verb put can imply putting something down gently or safely, whereas sending a person to the moon says nothing about getting the astronaut back safely.