These are not connected thoughts. Someone who saved a cat stuck in a tree, who is also a murderer, is still a cat-saving hero. The murderous act does not invalidate the cat heroism.
No, but if the mayor planned to honour him in a speech on national television, it would be common sense to include something to the point of, "Despite coming from a rocky beginnings, he has made great strides and today, is a hero." in order to specifically recognize the act of valor and not anything else. Or, even more basic, pick someone who you know isn't a murder who achived something similar to honour. These aren't exactly complex ideas. But again we didn't even get that far, its common sense to do the research to even check if the person has a criminal background or something similar that would reflect badly on you to ignore (nonetheless praise). They couldn't even do that much. Even if you think it shouldn't reflect badly on them, it very obviously will, and it doesn't take a expect to know that praising a Waffen-SS member as a hero does not look good. Again, if they thought that was okay to praise him, why all the apologies and the resignation.
The representatives' job is to represent the people, not to make decisions for the people.
Yes, this is why they just sit there after we vote for them. They have no impact or imput on national policy or law whatsoever and should not be expected to have any knowledge on such matters. Its not like they're representing us in a goverment or anything.