Yes. Generally, in North Welsh, 'Sut' is pronounced as in 'sit'. In South Welsh, 'Sut' is pronounced as in 'shut' - however, colloquially, sometimes the 't' is left out. So in this case, I would pronounce it as how you suggested, 'shu dych chi?'
This is also seen in pronouncing hello: NW - "Sut mae!' and SW - "Shwmae!'
Hope that's helpful :)
It's from bod.
Wiktionary lists the forms ych and dach; Gareth King, in his Comprehensive Grammar, lists those when he discusses North/South forms of the present tense (§ 227: ych in the south, dach in the north) but gives the single for dych in the "partially simplified overview" of the verb as a whole (§ 226).
He also notes that "Some tenses of bod, particularly the present and imperfect, show not only considerable regional variation, but also drastic divergence (in many cases simplification) from the ‘underlying forms’".
Welsh does not have one single standardised written form and so some words - perhaps bod even more so than others - comes in a wide variety of dialectal forms, many of which you may see in writing.
Some sources use some forms, some use others.
The "standard" form chosen by the Duolingo course is dych. The more northern form dach should generally be accepted as well but is usually not shown in "translate from the Welsh" sentences.
ydych is the form you most often see in learning material: https://en.wikibooks.org/wiki/Welsh/Grammar/Verbs/Present#Interrogative
The first letter is often dropped in casual Welsh (although ydych is often used too!), making it the same as the non-question form dych (itself a shaved down version of rydych).
Where possible, Duolingo prefers to teach the simplest forms of verbs. The forms Duolingo teaches should be present in a table in the notes (accessible via PC, but not the mobile app).
And rydych is itself a shaved-down version of a yet longer and more formal form :) (yr ydych chwi or something like that.)
As I understand it, rydych is a standardised but theoretical form - found in textbooks but not itself one of the many traditional forms descending from the same formal ancestor as rydych.
That's my impression too. Although there are plenty of Welsh speakers who do say it, possibly to deliberately adhere to a standardised form, like what people do in a lot of other languages.
But bod varies a lot by dialect anyway - so I'm not overly keen on saying anything for definite!