If I understand your question correctly, there are 2 challenges in your data
1000 missing value for question 1. There is a whole literature on how to handle missing values in regression. Statistical Analysis with Missing Data by Little and Rubin is a very good reference for this topic. I think this is a minor issue for you.
conditional branching in survey design. I think this is your main problem. I am not sure if there is any statistical model that can handle this. What I usually do in practice is to combine multiple questions. For example, suppose your outcome $Y$ is a numerical scale of depression level and you have the following questions:
a) What is your gender?
b) If female, do you have a new born at home?
Let's say a mother with a new born is often associated with higher chance of depression. What you really need is a binary variable, which equals 1 if the respondent is a mother with new borns and 0 otherwise. However, you don't want to directly ask this question as you will lose many other useful information such as gender(and it is really a strange question and will confuse many!).
So what I would suggest you to do is to go through your survey and think about defining explanatory variables that have direct causal relationship with your outcome. This is often called data pre-processing, sth. I consider far more useful than using complicated statistical models.