You don't care if it is not delivering full rated power or if it is bad when hot because your system pulls less than 200W total and will never cause those conditions. You just to check to make sure your psu is not one of the PSUs that burns up or that gives the wrong voltage.
The 4160 and the 1050 will work fine together. However IMO you are likely better bumping up to the 1050 Ti for that little extra juice. Your PSU (Assuming it is a good quality one) will have enough power for it. Something like an RX 460 would also be available for around the same price I believe.
Some would argue you could even bump up to the 1060 for some applications, but with the GPU market at the moment, the 1060 is likely not worth it.
Google $500 gaming builds. Most are based on an i3. Look at their choice of video cards. They will tend to spec a stronger video card.
1050 is a good card, and is below the area where Cryptocurreny has really messed up pricing for video cards. Hard to see where this could go wrong. The only possible fail is deciding in a year that your frame rates are too low. But buying the 1050 now and having new options in a year or two after the cryptocurrency boom/bust cycle runs it course could be the optimal solution.
If you have a single special game you play then see if AMD or NVIDIA works better for that game. Some games are optimized for one or the other.