Yes, connect one of the 6+2 cables to your 1070's 8-pin connector and you can connect another 6 pin cable directly to the riser board, which is actually the recommended method. Throw those SATA power adapters in the trash as it will save you a lot of headaches down the road.
As far as using the Pico adapter, if you convert a 8-pin PCIe cable to power the EPS connector separately you can get by with the lower rated Pico adapter since you will be splitting the power draw across two paths. You can't use a PCIe power cord directly to feed the EPS connector, as the pins are reversed on the end that connects to the motherboard, meaning the 12V and ground wires are switched around.