Data Science Noob - Customer Scoring based on conversion probability
Your question could be closed for too broad, but let's give a try. You want the enrolment probability, this sounds like a logistic regression for me. Neither categorical nor continuous data type should present a problem. You may model the additional information, applications who don't have one simply be assigned a NA category. You can include the NA category in your model.

I recommend you read the book Applied Predictive Modelling, I think it has a section on credit-card applications, which is close to what you are doing. You should try to learn from it.