I searched a lot over the Internet and tried all the advice I found.
I have installed drivers from the ASUS site and changed
C:\Windows\inf\wpdmtp.inf so it now contains
; Nexus 7 %GenericMTP.DeviceDesc%=MTP, USB\MS_COMP_MTP %GenericMTP.DeviceDesc%=MTP, USB\VID_18D1&PID_4E42&MI_00
My device is in MTP mode.
Computer sees it in PTP mode.
USB debugging is on.
It was said that I need Windows Media Player 11, but I have Windows 7 so it should be included, although I don't know where to check the version.
Here is what I see in device manager after all failed attempts: