Effect size applies to the effect you are measuring. As the examples you put in your question demonstrate, even within the field of churn, there are different effects that are interesting and the way to measure them will vary.
As to small, medium and large, Cohen just came up with some guidelines to use when you don't know what to do; I believe he did this ...
No, your understanding doesn't seem to be correct. There are two parts to the paper: user clustering, and then fast response churn prediction. K-means clustering appears in the former but not the latter.
The idea is to pass each user's activity embedding through K different LSTMs, each of which computes a different user behavior embedding. The user ...
If you have the end-date for all customers, then as far as I can see, you don't have censoring. Hence, you could treat the time a continuous variable (possibly consider a transformation, such as the log if it has a non-symmetric distribution), and use a linear regression model.
In a business-analytic context, this is often referred to as churn
modeling or churn prediction. It employs a wide variety of statistical
ideas and methodologies including survival analysis, Markov-chain and
stochastic-process theory, hierarchical Bayes models, etc.