1. which type partition of AD hold by child DC as compare to Child DC: Child
DC - I assume you meant here a domain controller in Child Domain. By default it will have Schema, configuration and
Child Domain partitions . Schema and configuration
are forest wide that means all domain controllers in a forest will have a copy of these partitions Domain partitions are domain wide .
You will also have application partition for DNS -
when AD integrated DNS is used there are two application partitions for DNS zones ForestDNSZones and DomainDNSZones.
2. Is there Parent DC users object exist in child domain its system state backup : If the Child domain controller is Global catalog then it will have partial attributes
( limited users data) from parent domain as well .
3. If the Child DC down then users of child dc can login : Users of child domain cannot be authenticated when DC is unavailable in child domain .
Note: Users can login to their machines with cached credentials for some time (10 time it used be)if you have not changed the settings .