Is Knapsack Feasibility and Knapsack with Unit Values NP-Hard? It is well known that the maximization problem of Knapsack if NP-hard. How about just finding a feasible solution i.e. objective is set to zero? 

And also if we set the values to all one (not the weights) in the objective of the Knapsack problem is it still NP-hard i.e. $\max v^Tx$ where $v=(1,1,1..,1)$?
Just finding a feasible solution is in $P$.  The empty set is a feasible solution.  It achieves an objective of zero.  So, the answer to "does a feasible solution exist?" is always "yes!" (to the extent that the question is even well-defined).  Similarly, the answer to "is any solution whose value is $\ge 0$?" is always "yes!".  Thus, those problems are not NP-hard.

When all values are one, there is also a trivial polynomial-time algorithm to find the optimal solution.  You should be able to find it yourself: just try a few examples, and you will quickly discover the trick.