I don’t think this happens nearly as often as you think. At least not in movies/shows made in the 50+years or so.
JRRT on water in his book The Silmarillion
But the other Ainur looked upon this habitation set within the vast
spaces of the World, which the Elves call Arda, the Earth; and their hearts
rejoiced in light, and their eyes beholding many colours were filled with
gladness; but because of the roaring of the sea they felt a great unquiet.
And they observed the winds and the air, and the matters of which Arda
was made, of iron and stone and silver and gold and many substances:
but of all these water they most greatly praised. And it is said by the Eldar
that in water there lives yet the echo of the Music of the Ainur more than
in any substance else that is in this Earth; and many of the Children of
Ilúvatar hearken still unsated to the voices of the Sea, and yet know not
for what they listen.
So yeah, water’s important. ;-D
Once you notice it, you notice it 
In the UK and Ireland it’s a genuine thing to be offered tea when there’s bad news. It’s definitely not a TV show thing.
If there isn’t a culturally-accepted drink to offer in times of stress then I guess offering a glass of water would make sense.
A cup of tea actually does help a little. It gives you something to do with your hands, and calms your breathing without being told to do something like breathe into a bag. And it’s a dose of normality. It also means the offerer is away for a minute or two, which can be good breathing space for everyone.
On TV shows though, I have seen it - can’t think of any specific examples, but it serves to send the offerer out of the room so that the detective-type can talk to the distressed person, or the offerer can be a detective-type who can then snoop around for a moment.
In the UK, hot SWEET tea used to be recommended for a person suffering any kind of shock.
Tropes are merely audiovisual cliches, and cliches are based in reality. Water is necessary for life; we are mostly made of water. So water has an enormous symbolic quality. Water is used as a metaphor, and water is actually offered in real life to people who seem like they need it. A thing can be metaphor and reality.
There was a memorable scene in EastEnders, where a daughter arrived with a face of doom just as her mother was making tea. The matriarch took one look at her face and said “Right, I’ll fetch the biscuits, shall I?”
And sure enough, whatever it was, was a real crisis.
And a M * A * S * H doctor had to discourage a nearby British outfit from giving tea to belly wound casualties.
Why is that one your “original question”, and not the four surrounding it? And can you see how we might not find it obvious you “already understood” the answers to the first and third question you posed?
I think you need to back up your original question, because I don’t think “give the character a glass of water” is that common of a trope.
No I think it is pretty common, but you don’t notice it unless you’re looking for it. I just watched an episode of a show tonight (Mr Mercedes) after reading this thread and since it was in my mind I was wondering if I’d see someone be offered a glass of water after something bad happened and they were! I couldn’t believe the one thing I saw directly after reading this actually did have that scene!
SEE?? 
If you’re Angelica Huston in Prizzi’s Honor, and you want to be passive-aggressive, you OFFER a glass of water and POUR a glass of vodka.
It’s not that common.
Well I just saw it happen again in another show. A guy was beat up and someone brought him a glass of water.
Haha I love this. Maybe we should do a public Google spreadsheet where we can track the actual frequency of this trope.
Wait till you hear my next one–it seems even less likely, but is possibly even more universal!
Yes by all means