[Chondrally (OP)]

Can we build a SynGas engine that emits no CO2 and is oxygen neutral?

It is possible to build a SynGas engine that emits no CO2 and is more fuel efficient than a traditional electric engine like a TESLA egine and consumes no oxygen NET!
SynGas is H2 and CO in the right proportion per volume.... it is a safe fuel if you mix it with enough Argon to make it less explosive but still combustible enough!  It is quite talk=sic, therefore a closed system with closed valve refuelling is necessary, and driverless cars will make it less likely, hopefully, if they do it right, to have far fewer accidents.  Care would need to be taken to insulate the passenger compartment from the fuel system in the event of an accident, and make sure the fuel system is unlikely to leak in the event of an accident.  Trying to make it as foolproof as possible however it wouldn't be impossible for a serious leak to occur in a severe accident, which hopefully would be very rare.
You can use Solar power to convert CO2 and H2O , the waste products of burning SynGas with O2 back into the fuel
again.
CO2 can be converted separately into CO and 0.5 O2 with a tungsten catalyst.
The H2O separately can be converted to H2 and 0.5 O2. 
The extra oxygen can be separated from the CO and H2 with a molecular sieve, or because of point of origin like with H2O, because you need two electrodes,  often made of platinum, but steel will suffice.
You need to mix the H2O with potassium (K) to form an electrolyte that will allow you to electrolyze it more efficiently.
When you remix the H2 and CO you have the fuel again.
The solar power can be used to electrolyze the H2O,  and a heat pump with heat from the waste products of the engine can be used to use a higher temperature to catalyze the CO2 into CO and O2.  These can be separated with a molecular sieve.  The oxygen should be released to the atmosphere by premixing it with enough air and exhausted to the atmosphere.  This engine is oxygen neutral and does not NET consume oxygen.
It is possible to build the engine without a molecular sieve,  if you mix the CO and .5O2 with Argon and air to the right proportion before injecting it into the combustion chamber to mix with the H2.  The H2 is naturally separated from the O2 because H2 comes physically from the negative electrode and O2 comes from the positive electrode thus they are pysically separated in space and can be kept separated, it is best to offgas the O2 from the electrolysis by premixing it with air or keep it to mix in with the CO and O2 and air (nitrogen and argon and O2) prior to injecting into the cylinders.

H2 does make metal brittle and corrodes it over time, so the H2 pump and fuel lines and H2 injectors will probably need to be replaced everty 3-10 oil changes.

The engine is practical and the O2 never has to exist with the fuel at the same time, unlike a methane cycle and so is much safer than a methane recycling engine.
It might be possible to get from 3000 to 8000 km on a single tank of 55Litres of SynGas.
We wouldn't need a war for Oil anymore! 
It might make a big revolution for Aviation,  as the range would be considerably extended and there would be enough
oxygen to make the cycle work even at high altitudes, so it might be possible for them to fly above the weather.
Check out:
https://today.uic.edu/breakthrough-solar-cell-captures-co2-and-sunlight-produces-burnable-fuel
https://www.scientificamerican.com/article/can-carbon-dioxide-replace-steam-to-generate-power/
https://www.theguardian.com/environment/2010/may/06/co2-green-fuel-car
http://newatlas.com/molecule-co2-carbon-neutral-fuel/48390/

 

Very important: The theoretical perfect efficiency of an engine is (T1-T2)/T1 where T1 is the temperature of the hotter reservoir and T2 is the temperature of the cooler reservoir.  This formula was probably discoverd by Lord Kelvin, but it was first described and directly followed from the work of Sadi Carnot:

https://en.wikipedia.org/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot

https://en.wikipedia.org/wiki/William_Thomson,_1st_Baron_Kelvin

A perpetual motion machine is actually impossible, as entropy and friction and losses exist, but we may be able to get almost infinitely close to it eventually!

https://en.wikipedia.org/wiki/Perpetual_motion

Time Crystals bypass the laws of themodynmaics themselves as they are not a closed system.

If we go to an open system, with access to energy from the Universe, it might be possible to have what seems like a perpetual motion machine to the uninitiated.   The energy has to come from somewhere, like the magnetic field of the earth.  If we draw energy from the magnetic field of the earth, and everybody does it, it might eventually deplete the magnetic field of the earth and stop it from protecting us from solar system Cosmic Rays.  Causing cancers and illness.  It might even not be safe to be around such a machine as strong fields can cause illness.

The engine described above is NOT a perpetual motion machine and needs refilling eventually, however it is more efficient than an traditional electric engine that does not draw energy from the magnetic field like the TESLA engine.  This engine is not a closed system, and does draw extra energy from solar power to do the electrolysis, and a 1 square metre solar panel is sufficient to accomplish this on a 4 passenger motor vehicle. The heat pump can be powered by the engine and a battery and an electric alternator.

[Bored chemist]

it is a safe fuel.

It is quite spectacularly poisonous.
Like all gaseous fuels it's potentially explosive when mixed with air.

[chiralSPO]

Syngas is extremely toxic, difficult to handle (hydrogen gas is soluble in many types of metal, including steel. This allows hydrogen to leak out, and it weakens the metal), and has extremely low energy per unit volume (even when compressed to 100 atm).

All of these problems can be resolved if we use syn gas for what it's meant to be used for. Synthesis of fuels. Syn gas can be converted into gasoline or diesel fuel, which is relatively safe, easy to transport, and store, and has very high energy content per unit volume. If the syngas is derived from a renewable carbon source (carbon neutral), then the fuel is just as useful or more useful on all counts.

If you want to go through the trouble of dealing with hydrogen, you may as well go all the way. Eliminate the toxicity cost of the CO, and remove carbon from the technology all together!

[Bored chemist]

Or use solar power, and a battery.