PLEASE HELP Belongs to vs falls under the / Just like /the way

Could you please take a look at this?

1 This district belongs to this agency/falls under this agency.
2 This sector of work belongs to/falls under the (authority of) minstry of...
3 The people living in this area belong to this school/fall under this school.
4 He trusts me just like I trust her./the way I trust her.
5 Just like its easy to get an apartment in the states, its easy to get evicted.
6 To get your dogs age in human years multiply how many years old the dog is by 7.

Thanks