Lowest cost build consists of lowest end H110 mobo + 8GB DDR4 RAM (standard value RAM) + 1TB HDD (no SSD) + entry level true power 500W PSU + cheapest variant of GTX1060 6GB/RX480 8GB/GTX1070 8GB.

For i5 6500 + GTX1060 6GB/RX480 8GB, it will cost around 3.3k+/-.

For i5 6500 + GTX1070, it will cost around 3.8k+/-
Yes, DSR is Nvidia's technology (AMD equivalent is VSR), you don't need a GSYNC monitor to enjoy this feature, activate it in NCP or NGE.

GTX1060 is a 1080p card so unless the games is really well-optimized, I wouldn't recommend you to use DSR (max out in games graphics settings instead of playing with higher resolution), GTX1070 (and higher) will benefit more from it on a FHD panel.

Yes, not included yet.

Imagine you have a 720p screen and a 1080p screen on 7" tablet, you will notice the display and text are much sharper on the 1080p screen thanks to its higher ppi (pixel per inch). This applies to PC monitor resolution as well, higher resolution will bring more details (pixel counts are higher), thus creating a sharper display.

GTX1060 6GB is best fit for 1080p, I wouldn't call it overkill unless you aren't planning to run games at maximum settings (or 80% max etc.)

Intel Core i5 6500 - RM 902
Asus H110M-K - RM 253
Corsair 8GB DDR4 2133MHz Value RAM - RM 229
Palit GTX1060 6GB GDDR5 Dual - RM 1229
Seagate Barracuda 1TB HDD - RM 219
FSP Hexa+ 550W - RM 188
Segotep Gank II - RM 179

Total: RM 3199 (ETech price inclusive of GST)

SSD is not included.
No, it depends on resolution, you get a higher ppi on smaller screen when resolution is the same, so the display and text are much clearer and sharper.

Bottleneck happens only when it cannot handle the amount of requests from other hardware, i5 6500 should be fine for most of the titles out there, if it really happens then you can just lower down the settings a bit to remove the burden of the CPU, that's one of the ways to handle CPU bottleneck, don't worry too much on the bottleneck issue, there isn't a standard hardware requirements for PC games (due to wide variety of hardware choices and game engine etc.) so just keep calm and game 
Your 7950 should be able to handle 1080p medium settings, if you really want to upgrade, please look for at least GTX1060.