In a small saucepan, combine the ginger with the vinegar, sugar and chiles. Bring to a boil, stirring to dissolve the sugar. Transfer to a large bowl; let cool.
In a large pot of boiling salted water, simmer the shrimp until pink and cooked through, about 2 minutes. Drain the shrimp and add them to the ginger mixture. Stir in the ice cubes and refrigerate for at least 4 hours or overnight. Drain the shrimp and pat dry. Discard the ginger and chiles.
In a small bowl, mix the Thai chili sauce with the lime juice, ketchup and fish sauce. Refrigerate until chilled. Arrange the shrimp on a platter with the celery sticks and cucumber spears and serve with the sauce.
Make Ahead: The marinating shrimp and the dipping sauce can be refrigerated separately overnight.