Ubuntu 14.04.1, Steam crashes on first login

I am trying to run Steam on Ubuntu, just double-upgraded from 13.04 to 14.04. There are other issues with Steam on 14.04 reported everywhere on the web, but I cannot find a similar one.

I tried both Ubuntu Store and the Steam Store versions and got into seemingly the same error.

I install Steam, agree with licensing and get to "setup account or login" prompt. When I login nothing happens. The Steam just does not load -- it crashes silently.

Here is the output in the terminal:

$ steam
Running Steam on ubuntu 14.04 32-bit
STEAM_RUNTIME is enabled automatically
Installing breakpad exception handler for appid(steam)/version(1416617579)
Installing breakpad exception handler for appid(steam)/version(1416617579)
Installing breakpad exception handler for appid(steam)/version(1416617579)
Gtk-Message: Failed to load module "overlay-scrollbar"
Gtk-Message: Failed to load module "unity-gtk-module"
Gtk-Message: Failed to load module "overlay-scrollbar"
Gtk-Message: Failed to load module "unity-gtk-module"
Installing breakpad exception handler for appid(steamwebhelper)/version(20141121162341)
Installing breakpad exception handler for appid(steamwebhelper)/version(1416587021)
[1229/213113:ERROR:nss_util.cc(1018)] Failed to load NSS libraries.
Installing breakpad exception handler for appid(steamwebhelper)/version(20141121162341)
Installing breakpad exception handler for appid(steamwebhelper)/version(1416587021)
Installing breakpad exception handler for appid(steamwebhelper)/version(1416587021)
[1229/213113:ERROR:nss_util.cc(853)] After loading Root Certs, loaded==false: NSS error code: -8018
Installing breakpad exception handler for appid(steam)/version(1416617579)
Fontconfig error: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 70: non-double matrix element
Fontconfig error: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 70: non-double matrix element
Fontconfig warning: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 78: saw unknown, expected number
Installing breakpad exception handler for appid(steam)/version(1416617579)
Installing breakpad exception handler for appid(steam)/version(1416617579)
Installing breakpad exception handler for appid(steam)/version(1416617579)
Installing breakpad exception handler for appid(steam)/version(1416617579)
Installing breakpad exception handler for appid(steam)/version(1416617579)
FillInMachineIDInfo took a total of 0 milliseconds
Installing breakpad exception handler for appid(steam)/version(1416617579)
Installing breakpad exception handler for appid(steam)/version(1416617579)
Uploading dump (out-of-process) [proxy '']
/tmp/dumps/assert_20141229213108_1.dmp
/home/alex/.local/share/Steam/steam.sh: line 729:  5820 Segmentation fault      (core dumped) $STEAM_DEBUGGER "$STEAMROOT/$STEAMEXEPATH" "$@"
Requested Force create but SharedObjectMutex already created
Forced create but already created for SharedObjectEvent
Forced create but already created for SharedObjectEvent
Finished uploading minidump (out-of-process): success = yes
response: CrashID=bp-b576bcbf-9b6f-44b6-9fcc-41b212141229

If somebody has had similar issue or just can help -- please post your suggestions.

Also, I run Intel's video card:

$ lspci | grep VGA
00:02.0 VGA compatible controller: Intel Corporation Core Processor Integrated Graphics Controller (rev 18)