Yes, your answers are required by law.
As you noted, the Constitution gives Congress and its delegees authority to conduct the decennial census "in such Manner as they shall by Law direct."
While it appears that you read that phrase narrowly to permit little more than headcounting, the courts uniformly reject that approach and read it very expansively, permitting inquiries into virtually anything the government wants.
The federal government's authority to conduct the interim American Communities Survey, which is more likely the survey you're taking about, is likewise unquestioned. Though there have been many challenges to many aspects of it, federal courts almost universally reject them, relying on the broad language of the Census Clause and the centuries of unbroken historical evidence demonstrating that everyone has always understood the government to have this power. The only successful challenge to a Census program that I can recall was Department of Commerce v. United States House of Representatives, 525 U.S. 316, 343 (1999), which held that the Census Bureau was collecting too little data.
More typically, people challenge the Bureau's authority to collect this or that data point, and they are uniformly turned away. See, for instance, United States v. Rickenbacker, 309 F.2d 462, (2d Cir. 1962), where the defendant was convicted of neglect to answer Census questions. Despite his argument that he could not be required to answer "relating to the contents, construction, and conveniences of the house or household in which he lives,” the trial court sentenced him to prison, fines and probation. The Second Circuit affirmed and rejected all his arguments:
The authority to gather reliable statistical data reasonably related to governmental purposes and functions is a necessity if modern government is to legislate intelligently and effectively. ... The questions contained in the household questionnaire related to important federal concerns, such as housing, labor, and health, and were not unduly broad or sweeping in their scope. The fact that some public opinion research experts might regard the size of the household questionnaire "sample" as larger than necessary to obtain an accurate result does not support a conclusion that the census was arbitrary or in violation of the Fourth Amendment.
