Thanks, Laz, seems finally I got it (till the next 4 years later when I forget it all). After doing in lab all possible scenarios I can seems discard all the contradictory Internet explanations that people provide while still confused with this matter :-). It actually sticks with laconic CISCO command reference and can be stated as:
‘network’ forces BGP to advertise EXISTING in RIB (no matter how it got into RIB) default to all neighbors in address-family and treat it as originated by BGP locally (where network command is entered) route;
‘default-information originate’ forces BGP to advertise EXISTING in RIB AND redistributed into BGP default to all neighbors in the address family and treat it as redistributed route;
‘neighbor default-originate’ forces BGP to suck out of thumb the default route and advertise it to particular neighbor unconditionally and treat it as locally generated route;
In addition, it is not recommended to use combination of these commands together. Considering subtle but important differences in functions and similarity in naming, no wonder this trio causes so much confusion. Particularly if to add OSPF similar but working a little differently commands.

As to the second question - sure, it was generated after several hours of headache in the lab addressing the first question. I realized the culprit a little later.
Thank you