Can an outlet be installed in the blank door panel in front of a sink?
It's no more wrong than an outlet in any permanent cabinetry.  Which is to say, not.  

However, it seems rather useless. Kitchen appliances have cords only 2 feet long.  I do not see how you could make any use of it without draping cords across the sink, the very thing Code seeks to avoid! 

As far as Code requirements to have kitchen counter receptacles within 24" of any counter space, this outlet won't meet that requirement brcause its use is impracticable.  If cutting a hole in the counter for a flat-mount receptacle does not appeal, I would use a pendant dropped down from the ceiling.  Pendants are awesome. 

As far as GFCI, do you really have to ask?  It is within 6 feet of a sink!  Of course the GFCI doesn't have to be right there; it can be a plain outlet protected by a GFCI somewhere else.