I think the "s" sound you're hearing comes from holding out the "d" at the end of "schild," as you then make the "pa" sound with your mouth after it. I tried it myself and by holding out the "d" sound (like the speaker does) I make the same "s" sound automatically as I transfer into "padden."
The final n is usually not pronounced in everyday usage - a bit like a native English speaker often dropping the final "g" in words. Eg "I'm goin' to the shop". Apparently it sounds a bit posh if you pronounce the final n. Not sure about the middle d though to be honest. I know it's often not pronounced in "goedemorgen", but don't know the details.
Rather, the final "-n" is diminished; it's not as prominent as in different positions. Though it is eroding, it's still possible to hear the remnant of the -n. How much of an "-n" you should pronounce is a matter of "damned if I do, damned if I don't". Not matching the pronounciation makes you stand out as an outsider, and not in a nice way.
I put "The turtles eat THE rice" and got a raspberry from Duo. How do I tell the difference in pronunciation between eten rijst and eten de rijst, given that Dutch often drops the final n from eten? Should I expect to hear "AYtuh duh RAYsht" for "eats the rice" and "AYTuh RAYsht" for "eats rice"?