Is a Graphic Designer expected to always do a "check spelling"?
No, it should not be a client expectation. Sometimes clients don't understand where the line between graphic design and copy writing is and they might assume that it is the designer's responsibility but I am always very explicit that copy editing and graphic design are two separate services.
For technical design assignments (Ex: Healthcare), Should a graphic designer be expected to look up/suggest corrections for organizations and/or industry terms?
No. I have worked in healthcare and finance for many years and we always proofed our design work with the client (together in the same room or on skype or goto meeting) before sending to print or publishing on-line. If your client is in an industry that has a specialized language, this is an absolutely necessary step in your workflow.
Also, I have worked with clients in many different countries laying things out in languages that I don't speak. Again, we always sit together and review the work with translators and other language experts before publishing. The big issue in working with other languages that I've encountered has to do with where the lines break, sometimes creating a poor reading rhythm.
Is it common for a freelance designer or agencies to include such clauses in their contract for grammar and spelling mistakes.
I think if they (freelance designers and agencies) are experienced, they will have clauses in their contracts that relieve them of the copy writing / editing responsibility.
If there is a spelling mistake, how have you informed your client of a potential spelling mistake (Is this too broad)?
I always run whatever copy is provided through a spell checker before putting into a layout. If I find mistakes, I will let the client know BEFORE doing any design work and ask for corrected copy or confirmation of the misspelling.