Check if knapsack problem instance is unsolvable
This is a complex set of constraints.  The standard way to solve this kind of problem is using integer linear programming (ILP).  Solving ILP instances is NP-hard -- so there is unlikely to be any polynomial-time algorithm -- but in practice, on real-world scheduling problems, solvers can often find a solution in a reasonable amount of time.  For your parameters (10 employees, 75 tasks, a few dozen time slots), I would expect that ILP might be very effective.  This kind of approach is widely used in operations research for scheduling.

So, use an ILP solver.  You can even set a time limit, and then it will return either "solution found", "no solution exists", or "time exceeded".  That will meet your requirement of identifying some instances that are unsolvable, efficiently.