It isn't necessarily required, but it does not have to be removed. It definitely does add meaning to the sentence.
You could put your bag on a table, in which case it would not necessarily be down. You could also put it into an overhead rack, in which case it would usually be up.
Similarly you can stretch your arms straight out, backwards, down, or just generally stretch them where they are.
It is possible to remove the preposition if you do not require the extra detail, but it is not necessary to do so: generally a reader would expect this kind of detail to give full context to what's happening. There is also the question of whether the sentence does need that context.
For example stretching requires no context: unless stretching their arms up is important for a specific exercise
Stretch your arms fully.
However without "down" then Emma's bag loses context. We can't say this:
Emma put her bag and went upstairs
Because you can't just "put" a bag. You have to "put" it somewhere. We'd have to replace "down" with something else, which would usually actually add un-necessary detail
Emma put her bag on the table and went upstairs.
Now we've brought the table into the sentence for no real reason. "Down" is a nice generic term that states Emma relieved herself of possession of the bag before continuing with the next activity.