Whether the internet will expect your server to resolve other "tier 3" domains
No, it won't.
DNS delegations act on subtrees, and normally they're all-or-nothing1: the administrators will not register your server for "all subdomains that we don't have" – they'll register it only for the specific level 3 names you asked for.
So if the parent domain's administrators delegate your server for xyz.⌷.com
, then your DNS server is allowed to provide responses for that specific level-3 domain and any of its subdomains (i.e. level-4 or deeper), but not for its sibling level-3 domains.
(The same server can host xyz.⌷.com
and abc.⌷.com
, but it needs a separate "registration" for each of those domain names.)
Delegations at other levels work exactly the same way.
1 (It is perhaps technically possible to set up a "wildcard" delegation, but nobody does that. RFC 4592 says "inclusion of wildcard NS RRSets in a zone is discouraged, but not barred.")
Whether you can use your server to resolve domains which it doesn't host
The short answer is "it depends on your configuration".
There are two different functions of DNS servers:
An authoritative server, which answers for specific domains directly from its database;
A recursive resolver, which provides answers based on what it receives from other servers. (It can talk to a single upstream resolver, or it can directly talk to authoritative servers.)
The same server can perform both functions: it can provide authoritative responses for domains X and Y, while providing recursive replies for everything else. This is somewhat common on intranets, and indeed your intranet DNS server sounds like it's performing both functions.
So if your DNS server directly hosts xyz.⌷.com
, that will not prevent it from being used to resolve the rest of ⌷.com
or any other domain – it'll just forward the queries "up" to the correct server.