is it more efficient to use a battery with an electric engine, to use a hydrogen cell to conver to electricity on an electric engine or to use a hydrogen combustion engine?
What is more important here is that each of them has unique advantages. These lead to them to function almost differently. For example, fuel cell (hydrogen) allows long distance travel even far from the base. Refueling is also fast. Electric & battery on its own may be good for short distance travel, but charging takes time.
More discussions are widely available in the internet nowadays.
It depends on how detailed the consideration is. A general statement with systems available on the market and their efficiencies:
Electricity --> transformer and rectifier --> electrolysis with ancillary units = total efficiency of approx. 65 %.
If hydrogen is now to be compressed and stored, the efficiency is further reduced depending on the gas pressure.
The conversion of chemical energy into mechanical energy means the use in a hydrogen engine? Please give me a concrete answer here, then I can also answer your second question accordingly.
The overall efficiency in the process chain electricity --> battery is more efficient compared to the electrochemical conversion, due to the lower number of process steps / conversion steps.
The use of battery vehicle or fuel cell vehicle can not only be done on the overall efficiency, but according to the application area. For daily driving distances of up to 150 km battery vehicles can be used (own estimation) and for daily driving distances of more than 150 km (own estimation) fuel cell vehicles are reasonable. At the same time, the choice between battery and fuel cell vehicles also depends on the vehicle weight. For example, it is advisable to choose a truck powered by a fuel cell rather than a battery.