I'll accept that there is the possibility of multiple speeds of slipstream, but isn't it therefore theoretically possible, with a fast-enough computer, to travel at *shudders* transwarp-salamander speed without actually turning into salamanders?
No, because that was infinite velocity. By definition, infinite speed is as far beyond slipstream's capability as it is beyond warp's capability. Increase your speed by a billion times and you're still exactly as far from infinite speed as you were before. (One of the many reasons why "Threshold" doesn't make sense.)
Besides, "Threshold" has been decanonized by its own writer. It officially never happened.
What was the drive in the TNG show they tested but could not get to work?
The soliton wave. A soliton is a self-sustaining wave, one that's constrained in a way that it doesn't dissipate, like a ripple in a canal or Jupiter's Great Red Spot. The idea in "New Ground" was to create a self-sustaining spacewarp that could carry a ship along with it at high speed.