Is the car on a flat surface? I mean, you're on a flat surface, can rock the car back and forth, can pull the shifter out of gear and into neutral but it still doesn't move side to side, the transmission is garbage. If the car's weight is not resting on the transmission, then you should be able to pull it out of gear. I think you would've noticed that if the tranny went out, though.
On the cable, is the connection to the firewall solid? Is the connection to the bellhousing also solid? If either is not, then your clutch cable may not be pulling the shift fork enough to disengage the flywheel from the clutch disk. Routing the cable isn't rocket surgery. Just keep the bend in the cable as gradual as possible and keep it off of anything that's too hot or that it would interfere with.