Is there a way to set up a column that must be completed if another column is updated in a certain way.
I.e. if you have a column "Is it approved?" which is set to "No" then the "Reason" column must be populated before saving the content type form?
I assume you are talking about Nintex Forms. Yes, this is possible. You have to use rules.
The community has a great page for you. Please check this page for more information about Nintex Forms.
The devil is in the details, so a little more input from me:
Scenario 1 also works in Nintex Mobile. Scenario 2 is more complex but gives you more options. This community site is full of useful information about this topic. The homepage of the community has a search box you can use for this.