Hello Vadim
Concerning point 1, indeed the network 0.0.0.0 and the default-information originate commands deliver similar results in that they will inject the default route into the BGP RIB, and it will be advertised to all BGP neighbors. The difference is in the origin of the default route. In each case, the default route comes from a different source.
With default-information originate, the default route is artificially generated and injected into the BGP RIB. A redistribution statement must also be configured to complete this configuration or the default route will not be advertised.
With the network 0.0.0.0 command, the default route will be injected into BGP only if the default route is currently present in the routing table. In case it is learned via redistribution, it must also be learned from the specific IGP from which redistribution into BGP is taking place.
The usage guidelines found within this Cisco Command Reference of the default-information originate command further describes the difference between these two commands:
Take a look at this post I shared with you four years ago about a similar discussion:
Concerning point 2, the reason you don’t see that default route readvertised to another iBGP peer is because of the iBGP split horizon rule which states “IBGP does not advertise prefixes from one IBGP neighbor to another IBGP neighbor.” It has nothing to do with the default route configuration but simply the way iBGP functions. More about that can be found here:
https://networklessons.com/bgp/internal-bgp-border-gateway-protocol-explained
I hope this has been helpful!
Laz