Shall I consider this behavior as a SW issue?
I should say so. You might even consider it a usability or accessibility issue. If something happens against what you expect you should file a bug. It can always be closed after investigation.
What is the best way to proceed?
The best way to proceed is to talk to your management and understand future development requirements, budget, business priorities, deadlines etc
Hiring a QA automation Lead to build FW and start implementing tests?
This is recommended only if your responsibilities and the new QA Leads responsibility doesn't overlap. But for ...
APIs are naturally part of SDKs. It is how you interface with code in libraries (jars, dlls, etc). When I worked on a library/package, I just had to write functional tests that used the library/package.
SDKs however are installable packages so there is also testing that goes along with that. When I worked on an installable toolbar we had to test the ...