How do I get unique listing of all folders on a Dropbox Business account.

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how am I guaranteed to get each and every folder that resides inside the team folder I am listing? I mean if the user 'Dropbox-API-Select-User' does not have permission to a particular folder inside a team folder, then apparently that folder will not be returned in listing call. Would using 'Dropbox-API-Select-Admin' solve this?

Yes, using 'Dropbox-API-Select-User' essentially lets you act as that user, so it would only have access to whatever that user has access to. Using 'Dropbox-API-Select-Admin' instead lets you access any team content. 

Also, when I list folders I am going to get duplicates. For example, member A and member B have access to a team folder called 'team_folder'. When I use /2/files/list_folder[/continue] with member A, I get the listing of all folders (including recursive listing of 'team_folder'). Then I use /2/files/list_folder[/continue] with member B and I will again get the listing of 'team_folder'. Am I wrong here? Moreover, if member A and member also have access to a shared folder, then I'll list that shared folder twice. 

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That's correct, if you list everything in multiple members accounts, and multiple members have the same team or shared folder mounted, you'll see the same entries multiple times. You can de-duplicate these by checking the "shared folder ID" or "team folder ID" in the metadata for the shared/team folders. Those IDs will be consistent across accounts.

Although, I have read through the doc that explains team space and non-team space, but I am not sure how would that affect my coz I am not checking/considering whether or not the account I am operating on has a team space.

The primary difference is that for teams not using the team space, the team folders are mounted inside each member's personal folder. For teams using the team space, it's reversed. I.e., the members' personal folders are mounted inside the team space. The latter would make listing everything a little easier, as the team folder(s) won't get duplicated across accounts.

Also, a shared folder can be mounted with different name in members' folders. How would I find the original name of that mounted folder and then generate a list of aliases (i.e. mount points to that folder in members' folder) 

You can get the name of a shared folder from /2/sharing/get_folder_metadata. You can list the mounted shared folders, including the paths (i.e., the mount points) in an account using /2/sharing/list_folders[/continue].