Q:

"My name is" or "My name's"

Can you contract the verb to be in this case? 
"My name's", "his name's", "her name's".... sound a little bit off to me, or at least sounds very, very informal. 
Most important question is "is it grammatically correct" or not.

A:

According to APA style, contractions are allowed but considered less formal. Your use of contraction is correct but not recommended in, say, an academic paper.
An excerpt from the above link:

Contractions are a part of informal writing. Thus, avoid contractions in scholarly writing, except for under the following circumstances:

If you are reproducing a direct quotation that contains a contraction
  (e.g., a quotation from a research participant), leave the contraction as-is. 
If you are writing about contractions (e.g., in a    paper about language), naturally you must be able to use contractions    as
  linguistic examples. 
If you are reproducing an idiom that contains    a contraction (e.g., “don’t count your chickens before they hatch”),    leave the
  contraction (no need for “do not count your chickens…”). 
If you are making an off-the-cuff or informal remark within an otherwise    formal paper, it is okay to use a contraction as part of
  your writing voice. You might find this kind of remark in a
  footnote or a parenthetical statement. Scientific writing should
  be formal but it doesn’t have to be stuffy. It is okay to have a
  moment of informality as long as the overall tone is appropriately
  formal.

