Todes-/death is indeed less formal than Hinrichtungs-/execution, but if you search in connection with a specific execution (e.g. McVeigh), all the words seem to be used pretty much interchangeably in both languages, except for 'execution room,' which is infrequent enough that I wouldn't include it. (Its meaning, like the rest of these actually, is clear from its parts.)
I agree with Oliver that 'chamber' does not necessarily connote gas; a gas chamber is just one type of death chamber.
To my ears Hinrichtungsstätte, -platz, -ort are more general terms, not necessarily indoors, and would be translated along the lines of 'execution area,' 'place of execution,' etc.
@Peter <de>: WordNet is just an online word list AFAIK; its definitions are sometimes helpful, but I doubt it's as reliable as good published dictionaries such as Collins, Oxford, Merriam-Webster and American Heritage, all available online.