Upvote if you think that this feature request should be implemented
Is this a feature-request worth asking about?
I think yes. There is nothing wrong in asking a feature request. And it seems like this feature request is good for Puzzling SE.
Is it too difficult to implement a system where questions can be awarded bounties as well?
I have no idea about it. But it was once said that "The amount of rep that is generated by each action (question up-votes, etc.) is too integral to the system to be a per-site setting.". But Stack Apps SE have +10 reputation for a single upvote on questions which is definitely different from other SE communities.
Are there any reasons why this should not be done?
If we talk about Puzzling SE, I don't think that there is any reason of why this should not be done. This feature will encourage good questions. It will also increase the participation on this site.
And should this be done on other sites as well?
I... don't really think that this needs to be implemented on all the communities. People ask question on other sites because of the some doubt in their mind. They seek answer to their problem. But it is not the case here on Puzzling SE. Questioner who posts puzzles here knows the answer to their puzzle in most of the cases. They aren't really gaining anything by posting puzzles here. They do it selflessly. In fact, in most of the good puzzles, questioner makes more effort in making the puzzle than the one who answers it. But there might be some other communities too who might benefit from this feature.
NOTE : This is a very short answer.