Ryzen 7 has 8 cores and should be the path for you to stream and game, as well as to edit videos. You will need 3000Mhz RAM to get the best out of Ryzen. Right now GPU prices are retarded, so a 1070 is all you can do for now. It will do good at 1080pand 1440p. All of that is here:
PCPartPicker part list:
https://pcpartpicker.com/list/MCJ48Y
Price breakdown by merchant: https://pcpartpicker.com/list/MCJ48Y/by_merchant/
CPU: AMD - Ryzen 7 1700 3.0GHz 8-Core Processor ($288.29 @ OutletPC)
Motherboard: MSI - B350 PC MATE ATX AM4 Motherboard ($55.98 @ Newegg)
Memory: Team - Dark 16GB (2 x 8GB) DDR4-3000 Memory ($165.75 @ Newegg)
Storage: SanDisk - SSD PLUS 240GB 2.5" Solid State Drive ($69.99 @ Amazon)
Storage: Western Digital - Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($46.49 @ OutletPC)
Video Card: EVGA - GeForce GTX 1070 8GB SC GAMING ACX 3.0 Black Edition Video Card ($619.49 @ OutletPC)
Case: NZXT - S340 (Black) ATX Mid Tower Case ($54.99 @ Newegg)
Power Supply: Corsair - RMx 650W 80+ Gold Certified Fully-Modular ATX Power Supply ($74.98 @ Newegg)
Total: $1375.96
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-02-15 21:58 EST-0500
All of this is below $1500.
Do you need a Monitor and Peripherals? For OS, you can download the ISO off Microsoft and install it but not activate if you do not mind an unactivated Windows.