Hi OP,
I'm so sorry to hear about this. i usually pop up as a cheerleader for kids with DS (mine is 7) and I totally stand by that, but I would never pretend it isn't hard work. I will start by apologising if any of what I'm about to say is way off-beam.
You have to throw a lot of the rule book for raising NT children out of the window with our kids. They have a different psychology and learning profile. Crack that, and a lot of the behavioural problems disappear. Have a look at this
Developmentally your niece is too young to be diagnosed with autism. I'm not saying she doesn't have it; she may well do; but developmental delay in children with DS means that an ASD diagnosis at age 5 would not be reliable. Please also be aware that many DS traits and behaviours can look like autism but actually come from a very different place psychologically.
No, she won't respond to consequences, reward or punishment. She doesn't have the understanding to anticipate 'If I do x then y will happen'. With the best will in the world, it's unfair and unrealistic to expect her to. This is where you do things differently. If my NT DC was refusing to get dressed or go to school I'd tell him what the consequence would be if he continued to misbehave, then I'd carry it through. With my DC who has DS I'd give him a big cuddle and (through gritted teeth) tell him how lovely he is. This will relax and distract him enough that he will then do what I want. It's counter-intuitive but it works. It also breaks the drama cycle. Young children with DS love a bit of emotional drama and reaction, and will engineer situations to achieve it. It doesn't matter whether that reaction is positive or negative. So if I spit food in a restaurant, Mummy and Daddy will be cross and tell me off, maybe take me out of the restaurant. I didn't enjoy the telling off but lovely drama! I'll do that again! Again counter-intuitively; ignore, no eye-contact, clear up quietly with no reaction or drama. She'll eventually get bored and stop.
That said, there may be sensory processing issues. Try to get her assessed by an occupational therapist specialising in sensory processing issues. We did and it was revelatory.
Our children are very emotionally intelligent, they have poor auditory and short-term memory, and they also have great fear of failure. This can mean they will try their hardest to avoid challenging situations, and NT people may not always recognise when a situation is challenging for them. One of my DC's less endearing habits is deliberately wetting himself (he is 95% toilet trained) when he's about to go into PE at school. He loves the activities but can't cope with the echoey quality of the loud noise in the hall during the lesson. Could there be something in her environment which is bothering her?
What support are your niece's parents getting? It sounds like maybe not much. There's a saying that behaviour is communication and that's acutely true for our children. Working out exactly what they are communicating can be challenging.
Does your niece have any speech? If not, do her parents or any of your family know Makaton signing? Do you think it's possible that she's frustrated because she can't communicate? Visual learning is a great strength for people with DS and many learn to read early. Could you teach her to hand you different flashcards with a mixture of words and pictures, for example a picture of a bed with the word 'tired', a picture of a heart with the word 'cuddle' or a picture of a cup with the word 'drink', etc. to express what she wants? If you repeat the words on the cards with her this will also help her speech. If you are in the UK your local childrens centre or Downs support group should be able to point you in the direction of Makaton lessons and support. Have a look at www.makaton.org/ and singinghands.co.uk/. Definitely buy her some of the Singing Hands stuff, I never met a kid who didn't love it as well as being a great therapy tool.
The toileting doesn't sound too out of the ordinary. Some children with DS can toilet train at age 2-3 but many are later, some even as late as 9. As with everything they have to learn it takes them longer to get the hang of it and they will often not be physically ready until later. It could be years unfortunately, it took us about three. She will get there but they key is to take any emotional load out of it. If people get frustrated with her when she wets or soils that could become a vicious circle. lots of praise when she gets it right of course.
I could go on and on and on, but probably more helpful if I just leave this here. It's a great website for anyone to get their head round how people with Downs tick and what we can do to help them reach their potential in all sorts of ways. I've linked to the US site but there is a UK version.
Good luck OP. These years are hard but all will be well.