Gentlemen,
I'm back from my trip; here are some [very, very belated...] random notes and impressions.
First of all, thank you again to everyone who replied on my questions! All your suggestions proved to be right on point!
Among the three cities, my personal rating is: 1) Genoa 2) Torino and then [a distant third] 3) Milano. Sorry to Milaneses over here -- and obviously, I'm speaking as a clueless tourist only -- but *way* too much people and the worst service in restaurants I had for a long time. Well, maybe it's because my visit directly preceded Milan's Fashion Week -- who knows?
One place in Milano is really terrific, though -- "Camparino" in Galleria Vittorio Emanuele II. Yep! -- it's probably in the most touristic spot in the whole city (Galleria on the one side; Duomo on another), but for me it's a coctail bar as it's meant to be.
"Antica Barberia Colla" is surprisingly non-modern (almost shabby) and down-to-earth establishment -- compared with, say, Trumper's in London, which is sleek and glitzy in comparison.
"Galleria d'Arte Moderna" is an interesting museum -- if you love late 19th / early 20th century art. Here you can see Italian artists you won't often find in galleries. In general, though, museums in these three cities is a big disappointment -- long queues, expensive admission tickets and guides generally available in Italian only!
Some notes on sartorial side. It seems every Italian city has one "British" (sometimes almost caricaturicaly British) shop.
Ghiglino in Genoa:
"By appointment to our best customers"!
There is "Sir Wilson" (sic!) in Torino. No photos, sorry.
Finollo is small, authentic and decidedly luxury store / atelier -- on the most luxe street in Genoa.
It has the reputation of being "the most expensive shirtmaker in Italy" -- so it was surprising to find that bespoke shirts here start from 250 euros only (but you have to pay extra for more expensive cloth, hand work, replacement collar, etc).
One unique offering is that they can make you a tie with a small image of your choice beautifully sown by hand -- and do so in a single day!
After some delibration, I decided to put my dog's tail along with my initials (in Cyrillic).
It seems such ties is a Genovese "thing" -- several shops offer them. Not sure who else can put a hand-made image, though.
There is a signed photo of DoW and Wallis Simpson on the wall
"...we made a shirt for him with this collar"
...and a shirt with suspiciously familiar initials...
Yep! -- Gianni Agnelli was also a customer.
"He ordered a very strange shirt, that combined a shirt with boxer shorts..."
Hmmm... Looks like GA took some sartorial hints (or misses?

) from DoW.
This is GA's collar.
And speaking on collars -- this is a "special" Finollo collar shape, along with a demonstration of its merits.
Andrey