Robots.txt File Editable?

When you say API  are you meaning a REST API?  If using the standard process for creating REST Apis they are already excluded in the robots.txt file

Robots.txt:

User-agent: *
Disallow: /*/tags/
Disallow: /tags/
Disallow: *.axd$
Disallow: /api/core/

URLs that are failing indexing:

If this is your file its already been modified.  There is no such thing as /api/core/  and there should have been an entry for Disallow: /api.ashx.  You are also missing several other important entries, I would go back to a default install package and get the the correct file.   

We are cloud hosted, should I put in a request with Support to compare what we currently have to the v12 default?

Yes

Yes