It was NOT "Via-Wizard-Installation" by the way. Because you can't install chrome from .deb package on bare Debian 9.1 easily.. So I had to "somehow" install it... That might be the root cause. So maybe I should close this one and rather open one related to .deb package not installing on bare Debian 9.1... But I'm lazy to do it.

[-omaha folks, since Linux install has nothing to do with omaha]

I'm not sure we're going to be able to debug/reproduce this without more details about what errors were seen or what steps were taken. It could be that however you attempted to install it caused apt/dpkg to end up in some messed up or partially configured state (and maybe would have happened with any package, but you just happened to be dealing with chrome at the time). The messed up state might then have confused gnome-software and been the cause of the crashes. Based on the lack of other such bug reports, it doesn't seem like there is any general Debian incompatibility.

Yes, we need to know how one "somehow" installs the Chrome .deb file before we can take any action here.

FYI, (recalling from memory) on a fresh machine, I usually do the following. Assuming of course, apt/dpkg is initially in a good state.
1) Download the Chrome .deb
2) sudo dpkg -i google-chrome.deb # It's going to be missing dependencies
3) sudo apt-get install -f # Let apt sort out all the dependencies

I have amd64 architecture, Debian x64 EFI
#1#: doubleclicking .deb defaults to: 
gnome-software --local-filename '/home/rules/Downloads/google-chrome-stable_current_amd64.deb' 
which always (even after fixing #2#) gives me: Screenshot
#2#: even when I installed GDebi pckg manager (GUI for dpkg as I understand) and opening google-chrome-stable_current_amd64.deb (see, 64! but still...) I got (same via bare dpk as I remember): 
"package architecture (i386) does not match system (amd64)" so I had to:
dpkg --add-architecture i-386
After that I could install from .deb 
Anyway:
1)why is that so with gnome-sowtware? (Note: there's no chrome in software-center's catalogue, only chromium)
2)why there are no "how to install via command line" on https://www.google.com/chrome/browser/desktop/index.html page where all required command will be written?
3) how do I after all install latest chrome 32 bit on my amd_64 laptop:
 installing google-chrome-stable_48.0.2564.116-1_i386.deb and opening gives me:
[1:1:0818/134708:ERROR:image_metadata_extractor.cc(112)] Couldn't load libexif. libexif.so.12: cannot open shared object file: No such file or directory
Gtk-Message: Failed to load module "atk-bridge"
Gtk-Message: Failed to load module "canberra-gtk-module"
Aborted

Thank you for providing more feedback. Adding requester "thestig@chromium.org" to the cc list and removing "Needs-Feedback" label.

For more details visit https://www.chromium.org/issue-tracking/autotriage - Your friendly Sheriffbot

Also, bug title can be changed to "Debian 9.1 Gnome-software"

thestig@,
Could you please respond on C#7 & update the thread accordingly.
Thanks..!

You're running "amd64 architecture, Debian x64", trying to install "google-chrome-stable_current_amd64.deb", and it's complaining about "package architecture (i386) does not match system"? That just doesn't add up. Assuming #1# and #2# are accurate, that error message must be about something else. Maybe from a previous attempt to install 32-bit Chrome (or some other 32-bit software)?

As for the other questions:

1) because Chromium is what distros build and distribute themselves, while Chrome only comes directly from Google.

2) because there's really not supposed to be anything tricky about it, and the basic commands that you've already tried, the same ones you would use to install any other package, typically "just works".

3) Why do you want to do that? 32-bit Chrome hasn't been produced in over a year, and even when it was, there was really no support for that configuration.

Also, why does that screenshot appear to be Windows? Is this all being done in the Windows 10 Linux subsystem? If so, I doubt that has ever been tested, and there could be any number of other things going wrong there.

It. wasn't Windows. It was set of tweaks to Gnome) Anyway I don't use Debian anymore. Switched to better Windows. And this issue may be closed or whatsoever. I don't care abt this anymore!