Hmm, this is a complex question but I'll answer as best I can. A lot of this will depend on you/your organization's risk tolerance and how much time they want to invest in tests. I believe in a lot of testing, but there is such a thing as too much.
A unit test tests the unit of code. Great, but what's a unit? This article is a pretty good discussion: http://martinfowler.com/bliki/UnitTest.html but a unit is basically the smallest testable part of your application.
Much literature (e.g. https://www.amazon.ca/Continuous-Delivery-Reliable-Deployment-Automation/dp/0321601912/ ) describes multiple phases of testing including unit tests which are very low level and mock externalities such as DBs or file systems or remote systems, and "api acceptance tests" (sometimes called integration tests although this is a vague term that can mean other things). This latter type fires up a test instance of your application, invokes APIs and asserts on responses.
The short answer is as follows: for unit tests, focus on the units (probably services or more granular), but the other set of tests you describe, wherein the test behaves like a client and invokes your api, are worthwhile too. My suggestion: do both, but don't call both unit tests.