One story involves a human who turns coat to help aliens in a war. The aliens with the traitor kill by hanging the entire population of a colony planet. This prompts humans to develop FTL travel to defeat aliens, as the "traitor" intended. Another involves either a colony or crash, only part I remember is one guy gets caught stealing food and they hang him. Yeah, a lot of nooses here. Read this collection once about fifteen years ago haven't been able to find it since.
The first story sounds to me like The Highest Treason by Randall Garrett. I had it in the ebook Randall Garrett Megapack, it was originally published in 1961, but I can't find the details as to where. No doubt it has been included in various magazines and collections over the years. Hopefully this can help you track down the collection it was in.
For reference, this is available at Project Gutenberg here