Thanks for the extra context. I will try to help, but remember, this is like doing surgery over the phone -- a bit dangerous.
Would you step in right away and coach/train the team or wait until later?
A story that does not meet the DoD is accepted by the PO.
Team decides to demo stuff that is not done in sprint review
During standups it's clear that some tasks drag on for days without much progress, noone in the team asks why
The first thing I usually consider is: What should my coaching stance be? See more on that here: http://www.agilecoachinginstitute.c...rt-one.pdf
What level is this team at? Shu, Ha, or Ri? (You can google that for more context)
If they are at Shu, then I might act a little stronger(Teaching, Mentoring stance) to ensure Scrum is followed. If they are at Ha or Ri, then I'm going to take a very laid back approach(Coaching/Facilitating stance) towards coaching the team into better decisions -- usually by asking powerful and socratic questions.
The next thing that I like to consider is "What the is the team's/org's sustainable pace in terms of accepting and adapting to change in their scrum implementation?" Then, if there are multiple challenges, I mentally or physically make a list, ordering from highest ROI to lowest ROI (value in terms of advantage to the company, over the political and real investment of effecting the change). Think of it as a "Scrum Master's" (or Coach's) backlog. You will also notice, as SM, that the Scrum Team probably has it's own "list" of things it things to be done. Anytime you can get synnergy between something on the SM list and the Scrum Team list, then that should immediately and dramatically increase the ROI of enacting that change. When in doubt, let the Scrum team try to remove that challenge themselves. You can help and facilitate, but let them try first.
I also consider strongly what will happen if I try ensure that Scrum and principles are followed, and whether team members might try to "go over my head" in order to not have to follow Scrum. I will often share the top 2-3 things on my "coach's" backlog with the person who is likely to be contacted by team members, and try to make sure that this person will back me up. I don't want to get fired for trying to do the right thing!
Keeping all of that in mind, it's time to ensure that Scrum is followed, albeit in a "servant leader" way. There might be a fair amount of resistance because it sounds like you, as the Scrum Master, have not been ensuring that Scrum is being followed for up to a year. As such, it may be wise to take a slow and cautious approach, maybe tackling one item at a time.
As an aside, when a person of authority is playing a Scrum role, or someone is playing more than one role, I always counsel them to make a lot of clarifying statements like "As the Scrum Master, my view is xxx" and/or "If I put my Dev Manager hat on, I think yyy" and/or "Speaking as a Dev Team member, I feel like we should zzz" I encourage those people to play those roles clearly. So, for instance, you might exert authority as a Dev Manager, but you should almost never exert authority as a SM -- because the SM is a servant leader. Hopefully you're also a servant leader Dev Manager, but you may be in an environment where that is not possible.
So, with my extra context, I ask you...
a) What do you think is the Scrum challenge that is happening on the team that has the biggest ROI right now?
b) How do you think you should approach that first challenge?
c) What do you think the Scrum Team would say is the biggest ROI Scrum(or non-Scrum) challenge right now? Have you asked them something like that?