Is there a practical carbon-free jet fuel?
A different fuel is not necessary. Even the most extreme experiment in diverted jet engine output, the Hunting H.126 did not divert its exhaust flow, it just tapped off compressed air. Its takeoff speed was 52 km/h. All aircraft need forward thrust, and with modern high-bypass turbofans you can get massive amounts of compressed air that's more than enough for blowing any surface you want. The BAe Harrier is a good example; the front nozzles of the Pegasus blow compressed air and match the thrust of the jet exhaust in the rear nozzles. 

Fuels roughly fall into 3 categories: 


hydrocarbons (everything from methane to jet fuel and diesel). They all contain carbon. Some leave less soot than others: methane is relatively clean. But it's a gas at room temperature, which makes it difficult to store (you need heavy high-pressure tanks).
Hydrogen. Clean, but very difficult to store (needing even heavier high-pressure tanks than methane), and explosive in a wide range of circumstances. 
Oddballs like hydrazine and hypergolics. They're all nasty, toxic fuels that require specialized handling which make them unsuitable for civilian use. Hypergolics are two-part fuels that spontaneously combust when in contact with their counterpart. 


Some more details on hydrazine: 


  Hydrazine is highly toxic and dangerously unstable 


Anyone involved in fueling a hydrazine-powered vehicle has to wear a full Hazmat suit. That alone makes it difficult to use a hydrazine-powered aircraft on most airfields as it'd disrupt normal operations. 

Hydrazine is also inefficient. For jet fuel, you can pull the oxidizer from the surrounding air. Hydrazine has a specific impulse of 220 s, while jet fuel has a specific impulse in the region of 350 s. So you have to carry 350/220 is 1.5 times more fuel to do the same work. 

The exhaust products of hydrazine are extremely flammable (hydrogen). You could burn that in a jet engine, which would significantly increase the specific impulse. Hydrogen is a bit finicky, but hydrogen-fueled aircraft have been tested.