MPLS VPN PE-CE OSPF Default Route

Two questions inspired by this lessons that maybe strictly speaking deserve its own lesson in BGP course:

  1. Rules for redistribution of default route into BGP - if it is from OSPF we need to explicitly inform BGP we want internal and external routes. From static routes it works straightforward. And maybe also from EIGRP (?). But thats not enough - we also need to confirm we want default that was distributed to be advertised by BGP. One way is to run default-information originate (and BGP does not have ‘always’ keyword, thats it will not advertise unless route is injected already into rib). But seems the same thing can be accomplished by running ‘network 0.0.0.0’ command. I tried it in the lab and could not find any difference, including that ‘network’ command does not do any good unless some other protocol had default already installed into rib. Only combination of redistribute default from OSPF (or static) and either ‘network 0.0.0.0’ or ‘default-information originate’ produces advertisement of 0.0.0.0 by BGP. Like say having ‘network 0.0.0.0’ and default-information originate in BGP without also redistribute of 0.0.0.0 from somewhere did not do any good. This is a strange logic to me but the main question is - is there indeed any difference between using ‘network 0.0.0.0’ and ‘default-information’ commands? (I don’t refer to ‘neighbor … default-originate’ - that one works fine without anything else but clearly not always convenient when you have many neighbors.
  2. Once default is in BGP and advertised to neighbor will it also be advertised to next neighbor? (assuming there are no explicit filters for it). It looks in my case it did not - the neighbor received default over BGP but did not advertise it to the nest neighbor (all three are iBGP neighbors - thats 1 and 2 neigbors and 2 and 3 are neighbors but not 1 and 3) . Does each neighbor need default-information command for default route to propagate? Would that behavior be different in case of eBGP?