Is it possible for Gross value added be negative on a company level? In National Accounts production approach, can there situations whereby the Output of a company is far lower than the Intermediate consumption therefore the Value added is negative?
Yes, it can happen if a company is selling things for less than it cost it to produce them, though for obvious reasons this doesn't happen very often (or at least not over any great length of time) as it usually doesn't make sense to continue producing in those circumstances.

You might see it occasionally, for example, in small farm businesses.  These can have high subsidies on production to begin with (which get subtracted in the calculation of GVA, narrowing the loss needed to bring it below zero), have few or no employees (just a self-employed farmer, so no or low fixed wages/CoE to be paid out), and a bad harvest or loss of livestock could result in the business not producing enough output to make GVA positive for a particular year.