I have a list which is used to study reading, listening, and tones but not writing. In the future I want to gradually study writing for the vocabulary on that list once I have become familiar with it. How could that be done? Keep in mind that this list is my Quick Add list so it would not be practical to change the list settings. Would I have to cut and paste the vocabulary to new a list? If so, would I lose the existing study settings?

Just make a Remix of this list, which makes a complete copy of the list.
Then you have 2 options:
1)restrict this list to writing only. Then you can slowly add the writing part. But what will happen in the future, when your quick add list grows?
2)allow all 4 modes, writing, reading, .. for this list and delete all items from your quick add list, because then they are already saved in the remix list. So you would always know, what new stuff you might have to add in the future for writing.

I'm also unsure about how this works. I have changed the study settings on one particular list that I only want to learn the definitions and tone. I only have these two criteria ticked but for some reason I'm getting asked to type the word using the keyboard (reading).

In fact it is still testing me on writing despite changing the study settings. Like I said I don't understand how this works. Should be more intuitive IMO.

shanghainese, is it still asking for writing for words that are added to My Words after you made the list-specific parts change, or just those that were already in My Words?

I'm having the same problem as shanghainese. It is asking for writing of words that are added to My Words after you made the list-specific parts change.

Basically I found a new list to study. Before I added any word from that list, I changed the study setting to tone and definition only. Then I start studying from this list. It is asking for writing of words from this new list that were not in any of my other lists.

Edit: I got it to work. The system seems to have taken a little time to process that parts change. I went back to studying the list and this time only the tone and definition parts were added to my words.