Does T-SQL have a Schema search path?
No. SQL Server will first check the user's default schema for unqualified object references, and then the dbo schema. If not found at this stage, then you get an error.
I've been searching high and low now, but I can't find a reference for this in the product documentation.  And it is even hard to find anybody written about this over the whole interweb. I did, however, find this from Bob.
Max Vernon wrote a post showing how this works here.
Aaron Bertrand has some interesting advice about always specifying the schema, here