Hard to believe but I'm still sketchy at identifying some orchestra instruments. 
 
What is the specific instrument which plays the slow melody in this video starting at 0:16? I figure it's a clarinet, but what type of clarinet exactly? 
 

Isn't it an oboe?!

Ah yes you might be right, the oboe has more of a "pinched" sound than the clarinet does it not....
 
How embarrassing.....I thought I'd at least whittled it down lol....

Definitely not a clarinet. I would agree with eph that it sounds either like an oboe or the lower register of a soprano sax.

Ohh I wouldn't have thought soprano sax though. It's an instrument that always stands out to me when I hear it and I think "that's really nice," and I keep meaning to properly identify it. It has a sort of pastoral quality which I like. 

Probably an oboe or the lower, closely related english horn. It could be sax, but I doubt it in this context. I remember being told that - if you don't have an oboe in your ensemble - the oboe sound can be approximated by a clarinet and flute playing the line together with the flute up an octave.

Well one thing's for sure. I'm surprised that nobody's come out with "oh it's definitely X without a shadow of a doubt" - kind of makes me feel better for not knowing!
 
At the minute, I'm going with Cor Anglais. Have a listen:
 

It's definitely "yanny".

it's double reed, she says it's more melancholy (the cor anglais)  In your excerpt the range the instrument is in is right at the crossover point for oboe.  Some of the higher notes are *pinched* like the oboe.  Do you know the answer?  i'M DYING TO NO!

Now I'm even more confused than ever....