how much can an old man improve at Blitz?

I'm 63 years old and my brain health is certainly good, relative to my age. But I surely can't think as fast as I could 20 or 30 years ago. I am respectably intermediate at Daily -- 1499 at this writing -- but to the extent I am successful it's because I work really hard, studying moves a long time and playing no more than 2 or 3 concurrent games. So I do OK at thinking slow and deep.

I enjoy the 5:00 games (sort of), and the adrenalin is a kick, but I suck at it and wonder how I can improve, other than by accumulating experience. If I think too much, I run low on time; if I think too little, I make mistakes. I know this has to be true of everyone, at least to some degree, but I am tremendously impressed by players who can think both fast and deep while staying sufficiently calm all the while.

The 5:00 games sometimes feel rather like a bad dream in which I am not able to see the whole board clearly, and I'm flailing rather than computing (sometimes the opponent flails and blunders as well, and there's plenty of sloppiness on both sides). I just wonder what would be a good way for me to train myself play better when playing fast.

There was a man in his late 60s who joined our club. He went from ~1000 strength to an OTB rating of 1400 before he quit. This was over a few years. His chess.com daily rating got as high as 1800 (this was almost 10 years ago, I don't know what the equivalent chess.com daily rating is today).

He didn't play much blitz, so I can't give you a rating for that, but it's certainly possible to improve no matter what age you are.

To be a good speed player, you need a foundation that comes from study and playing long games, but if your limiting factor is speed, my only advice is to play many 1000s of games.

It depends on time you have to devote and potential left to use. btw I love your profile pic!

There are a few ways to help you.

1. Use a touchscreen with a large area so you can see and avoid misclicks.  If you can't afford a large touchscreen, a mouse is better than a trackpad. 

2.  Practice your coordinates in Learn --> Vision, (Learn ->coordinates on lichess).  A lot of speed chess skill comes from just being able to manipulate the pieces fast. 

3. Do easy tactics as fast as possible as part of your training.  Like, do a bunch of mate in one's as fast as humanly possible.  Chesstempo.com has a blitz tactic setting that grades you not only on correctness, but also speed. 

I'm at your age.

Blitz is just for fun, so if it pleases you, just play and don't bother about your blitz rating.

Simple as that.

and what is classical chess "for" ?

In athletics 100 m sprinters are young, but 42 km marathon runners are elder.
Forget 5|0, play 5|3 or even better 15|10.

According to Fischer, it's for crushing the other player's ego.

Well, I agree with llama47, arnaut10 and optimissed that it is not much a matter of age, so the real question should be "how can a player get better at blitz?". Usually a player has quite the same rating in blitz, rapid and standard time control, but in your case the difference seems far bigger. You yourself provide some hints, though:

"I am tremendously impressed by players who can think both fast and deep while staying sufficiently calm all the while".

Maybe you are just a bit emotional and inexperienced in playing blitz? In that case probably llama47 is right: you just ought to play some thousand of blitz games and get used to them. In fact, blitz and standard are almost two different things: Anish Giri said once that in blitz you have not to play well, you have just to survive, so time management is paramount. The other advices might improve your play in general, too, so you can try and see. We all are interested in how to get better at any age, so best of luck!

I don't know if he meant it this way, but I often think the Hippocratic Oath is also a good blitz motto:  "first, do no harm."

What I mean is, find moves that are solid and that don't risk a lot. Even if your position becomes worse, it will take many moves to beat you, and you can win on time.

Yes, I think that's what he meant, very pragmatic.

Improvement.

If you are healthy, you can improve but it will be slow. That's what I believe. However, people usually improve less the more they grow older. 

So you better keep patience. I would not recommend blitz or bullet time control to use it as a form chess exercise to improve. I recommend slower time controls. 

In a book Boris Gulko wrote.  A friend of his that is some sorta brain doctor or something like that said that as we age we start to lose the ability for tactics, though strategy stays the same.  That is why we get worse as we get older. 

It's an interesting question.

I think the answer I'd give is... the longer the time control, the more the objective evaluation matters.

That's not to say blitz takes less skill. In blitz there are many practical factors that don't exist in classical OTB chess.

Life teaches us that we start at the beginning. 

When we are babies we learn to crawl, then walk, then run. 

We start with 2+2 = 4, not advanced/applied mathematics.

We learn to read by sounding out simple 1 syllable words, not speed reading War and Peace. 

Nothing wrong with playing speed chess, but if you haven't got a firm foundation of chess fundamentals, you wont go far just playing fast.

It's the same for G/60 or G/120 right? In study you spend a lot more time on positions than you do in tournaments. For those two time controls your average minutes per move is 1.5 and 3. This is far too fast to learn only by playing. In study you might spend 10, 20, 30 etc minutes on a single position. And you might come back to it on the next day, doing more work. You might explore many variations and make many notes. In all you can spend many hours.

So it's not so much that you have to learn classical before learning blitz, but that learning to play chess in general is required for any time control.

     You don't say how long you have been playing chess, whether you played when you were young, whether you quit playing for years and just started up again, or whatever.

     If you were a decent player who quit or hardly ever played for some years and just started up again, you are highly unlikely to ever be as good as you were. We old-timers lose a bit with age.

     If you took up the game recently, you're doing remarkably well. Continue to play and pick up a little "book learning" and you will continue to improve.

     The more you learn and practice all types of chess, the stronger your overall game will be. But realize that some players are much worse at blitz than at longer time controls because the more they can examine the position, the more possibilities they see and they find the strong moves more often.

     And as you enjoy the blitz games, keep playing them occasionally and don't worry about winning and rating.