Predicting dropout in an ordered process: Cox regression, autoregressive model, multilevel modeling?

Question

I am working on a project in which I collected data about 100 people’s steps in an ordered process. All took at least one step, with some continuing up to a fourth step. Each person either drops out after each step or continues on to an additional step. What each step actually consists of is different for each person. I have extensive data that describes the people and about their experiences at each step. Thus, the data are nested: descriptions of steps are nested within people. Here’s a very simplified example of my data structure:

I'm hoping to use independent variables at the level of the person / level 2 (e.g., gender) and at the level of the step / level 1 (e.g., the quality of their experiences at each step) to predict whether people persist in or drop out of the process after each step (dichotomous outcome- yes/no). For example, do people with bad experiences at a step tend to drop out of the process after that step? Are women more likely to drop out than men?

I originally set this up as a multilevel logistic regression in SAS to account for the nonindependence of the data but also tried running it as a cox proportional hazards model in R using the survival package to account for the fact that the steps in the process are ordered. The graphs / tests of the Schoenfeld residuals do not suggest that my independent variables are step-dependent, but there are so few steps and so few observations at step 4 in particular that I’m not sure whether to trust this. I did a bit more investigating and found the coxme package, which I’ve been trying to set up (so far, without success) to test a random effects cox model.

I’ve spoken to a few stats consultants and have gotten some mixed feedback about the type of model that I should use. Most recently, someone suggested that I try running an autoregressive model with a logit function, but I am totally unfamiliar with this approach (other than knowing that a logit function is the inverse of a logistic function and autoregression would take past steps into account) or how to implement it with my question and haven’t gotten much clarity from an initial search about it.

My questions are:

1. Is multilevel logistic regression, an autoregressive model, cox regression, or something else the best fit to my question?
2. Should I be concerned about the small number of observations when testing the step dependence of my independent variables?
3. If I were to use cox regression, would I need to specify a random effects model to account for the nonindependence of the steps?

I'm new to survival analysis but not new to multilevel modeling. I have some stats background but do not consider myself to be particularly strong in it, so I would appreciate explanations that take that into account.