...since the YN 560 IV can control other wireless flashes can it also control and group the [Neewer] TT560 flash since it's connected to a wireless receiver now?
No. It cannot; all you can do is fire the flash in sync.
The RF-603 II C, when used as a receiver cannot do power or group control, so there's no way to communicate that to the flash. An RF-605 transceiver can give group on/off control, but not remote power control.
In addition, the Neewer TT560 (which, like the Amazon Basics flash, I suspect is actually a rebranded Godox TT560) is a single-pin flash without any built-in radio triggering. Anything attached to its foot can only communicate sync (fire). No other signals can be received by the flash because it doesn't have the pins to receive them. Your YN-560IV would behave the same on an RF-603 II; it can only have its group/power controlled via its internal radio transceiver.
If you want remote group/power control, you have to stick with the YN-560III/IV/-660/-560Li models, which also have a compatible built-in transceiver.