I created one character without any problem, and when logging into the game the window popped up where you can choose whether or not to have the Auction House, Grouping, etc.
But when I created a second character this window did not pop up, I was straight into the game without being allowed to make these choices.
I deleted this second character and tried again. Same problem.
Is there any way to make these choices once the game has started?
Is this a known issue?
Thank you for any help 
This is not a feature offered by Official Hardcore, but by addons. Please check whether if your addons are up to date and are enabled.
Official Hardcore has trade and AH enabled from the get-go with no option to turn this off.
As green poster said this is not official feature, but of addon called Hardcore. That addon must have saved your options, but, since you are most likely using same name for consequent characters it might just used old files in WTF folder, because these are not deleted after you delete a character.
Did your character have same name as the first character?
Addons store their settings under the character name and has no knowledge if you delete and restart - thus you will inherit settings from the previous character with the same name.