Finally I got the solution for this case - which is just to wait... sounds strange, I know.. I just let the phone in installation process for a night and when I woke up it was saying that installation had proccedeed, but requiered my confirmation for downloading

And then I encountered my previously found bug which is unability to download any packages because it gets stuck at the beginning. But still, I decided to just press retry when it was giving me "failed to download, try again?" and after about 6 times of retrying it worked!! Pheewwww

Now I have a new problem, but I believe it`s not that rare - during an update of Maemo Community SSU Package it says that I need to use PC Suite, but as far as I understood it`s rubbish and the solution is somewhere else

Most of the time this is a free space, a lack of free space, issue.
Try to give some information about the partitioning and free space of your device by using the following code to create some file called debug.txt, that you might to attach to your next post here.

It gave me several errors after du -kx operation:

du: can`t open '/etc/ssl/private' : Permission denied
du: can`t open '/var/lib/pulse' ; Permission denied
du: can`t open '/root/ .pulse' : Permission denied
du: can`t open '/root/ .gnome2' : Permission denied

And also after sfdick command it wrote:

-sh: sfdisk: not found

I suppose the last one is about SD card, so I don`t have it
Debug is in attachment

The command is "df -h". Then look out for /

Thanks! Now I know how to check it
Also, it`s already been written in debug, so my free rootfs memory is 98.1/227.9 Mb (56%)

Is that enough to instal CSSU?

Thats enough space to install cssu!
There might be some other error which Nokia's software is incapable to debug, aka. garbage in, garbage out!
Which brings me to the error in my code above This might be the corrected part:Nevertheless, did you try to get the cssu installed via Faster Application Manager or in X-Terminal? This might be the sufficient code to install itPlease copy and paste the complete error message here.

I tried to do it through X Terminal, but it gives me errors about not being root, as far as I understood I need to install rootsh to do it, but when I try to install it through standard app manager it says:

As for FAM.. it`s so buggy, it takes long time to update, but in the end it still says failed to update catalogues or some other errors, however internet connection is fine and web browsing is fast

I misinterpreted your earlier post about some success downloading as a successful installation of rootsh.
Is the extras repository in Hildon Application Manager enabled?
Did you as first step in Faster Application Manager import the HAM repositories through 'Manage Repositories', then click on the title bar and 'Re-import HAM repo settings' ?
Did you look for the error log in HAM through titlebar menu?

EDIT:
You might get rootsh by clicking on the install button here http://maemo.org/downloads/product/Maemo5/rootsh/
or by downloading rootsh from the repository by browsing to http://repository.maemo.org/extras/p..._1.5_armel.deb, save the file to, may be Documents and click it in the in-built file manager.

Thank you for that link! I finally managed to install rootsh. However when I input apt-get update in X Terminal it gets stuck at 0% during updating repository.maemo.org and then lots of "Failed to fetch errors" and in the end:
What can be the problem now?

This is a connection error, like you have no internet access, or the servers are down. Please try to re-think your connection. Router, proxy, wlan, umts, firewalls during shared internet through a PC. This is very odd, that anything related to application download is blocked or times out.