How can I recreate the dipping sauce for the cream cheese wontons served at Pick Up Stix?
Can't help you with this sauce in particular, but I can give you how I would approach it and maybe that will help you.

First, have some of the original around so that you can test against it.  Testing against your memory will never work, although you may get some great results anyway.  It just won't match the original.

Second, take good notes of each attempt.  You're going to be working in one of two ways to create the recipe, either method requiring excellent notes if you are going to duplicate your success once you achieve it.

Third step you're doing right.  What are the predominant flavors?  What is the base, most likely?  In this case you've identified soy as the base, and then sweet, acid and ginger.  Ginger is fine (although you will have to decide if it tastes like fresh or ground), but the others are too vague.  What could the sweet and the acid be?  When you are trying to duplicate something, think of the cuisine.  In this case we are talking Asian (although commercial Asian) so the sweet could be Mirin and the acid is most likely rice wine vinegar which is a mild vinegar.  The Mirin would be a sure thing if this were not commercial Asian...odds are it is just white sugar and it only seems mild because the vinegar is mild.

Now you get to choose which method you want to work with.  You can make several small batches, changing things with each batch, or you can make a single progressive batch, modifying as you go.  If this was a baked good, you'd have no choice but to do each batch separately, but since this is a dipping sauce, you can modify a base batch slightly to get to where you want to be, and is how I would approach this.

Start with your base (soy sauce) and pick an amount that is reasonable.  Since this is a dipping sauce I'd probably start with a quarter cup.  Not too much, but not so little that I have to measure 1/16th of a teaspoon of the ginger, say.  Taste the original sauce, taste the soy sauce.  If the soy sauce is WAY stronger than the original sauce, they probably added some water to it.  If needed, add water to your experiment, measuring and writing down each addition, until it seems a little stronger than the original. 

Now go for the next major taste, probably the vinegar.  Add a little at a time (measure, write down) until the basic acidity is right, maybe a little shy.  Now balance the acid with the sugar (or Mirin) a little at a time, tasting yours and the original until they are close.  

Now add your top notes (ginger and sesame oil).  You should at this point be close, and can adjust.  If you put in too much of something (say the ginger) you will need to add a proportional amount of everything else to bring it back to balance.  This is why you wrote EVERYTHING down...you did, didn't you?

In the end you should have a batch larger than you wanted, but you'll know everything that went into it.  Look at your actual amounts.  Readjust all of the amounts to be something that you might actually make.  If, for instance, you needed 1/8th of a teaspoon of grated ginger to make this taste right and you know you'll never get that right again, multiply everything by four, so you can do your ginger at 1/2 teaspoon.  It's all about proportions at this point.

If, by the way, you wanted to do this as separate batches, you'd do the first batch as described above.  Decide what is off (too much sesame oil? too much vinegar?) and rewrite the amounts the way you think the next batch should be and then make it.  

By the way, if you believe that you didn't have enough of an item, you can always add it to the current batch to see if you were right.  Not enough? Try adding it to the current batch.  Too much? Either adjust the amount for the next batch OR using the progressive method above, adjust the other ingredients to bring things into balance and then proceed.

Note that in all cases I add less than I think I needed to get close, then sneak up on the amount.  Once you go past any amount, you have to do a more major adjustment.

And measure and write.  You will NOT remember the amount.  Really.

And, finally, have fun.  You may come up with something even better.