Monopoly/Monopolistic Competition Productively Efficient or Inefficient? I know both market structures are allocatively inefficient as P > MC. But,

Since monopolistic competitive firms produce on the downward sloping part of their AC curves, there is excess capacity which implies productive inefficiency. To achieve productive efficiency, they have to produce on the lowest point of their ATC curve. Since monopolies also do not operate on this lowest point of their AC, they are also productively inefficient. 

However, the total cost curve shows the least cost method of producing each output level as it is derived from the tangent of the isocost to isoquant, which implies that all points on the total cost curve is productively efficient since cost is minimised. Since AC = TC/Q, it also implies that all points on the AC curve is productively efficient - all points on the LRAC are productively efficient. 

Since all points on the total cost curve and average cost curves are productively efficient, monopolies and monopolistic competitions, who operate on their AC curves, are productively efficient.

Both of this concepts contradict one another. One claims both are inefficient while the other says they are efficient. Am i confused or is one of them wrong?

If productive efficiency is achieved when goods are produced at the lowest cost possible, are all points on the LRAC are productively efficient since the LRAC shows the lowest cost of producing at each output?

Productively efficiency is only achieved on the lowest point of LRAC/SRAC?
No contradiction. All points in the AC curve indeed reflect the production of the corresponding quantity at minimum cost. This is conditional efficiency, conditional on arbitrarily specifying an output level.  

Then we ask: what is the output level for which this product is produced at an average cost that it is lower than the average cost for all other output levels, the minimum minimorun, the least of all minima?

And we get the minimum of the Average Cost curve. At this output level we cannot do better by varying the quantity (either increase it or decrease it). So it is this quantity that achieves "universal" efficiency.