General approach on time series for customer retention/churn in retail
OK, let me try to dissect your questions:


  Q1: Should I be grouping customers like I am doing, on a monthly basis
  or I have to group them on a 2-month basis since that is how they were
  labeled?


This depends a bit on what your goal is and what kind of features you want to use. I think, with the limited things I know from your question, I would go for a daily prognosis for the next two months (so if you're doing daily you can forecast a period of two months daily data for each day). Reason being that otherwise you might loose some inactivity "features" that are implicitly part of your data. As an alternative you can group your data and create features that help you introduce the information anyway. For example: activity_in_last_5d, activity_in_last_10d, etc.


  Q2: Also, how do I model this? Do I keep customer_id as a feature of
  the model or not? 


Customer_id doesn't seem to be a feature to me but rather the key for identifying which features to feed into your model.


  Is the gap in dates for each customer relevant and
  how should I deal with it (if)? 


You are basically asking about imputing your data (see this for more: https://towardsdatascience.com/6-different-ways-to-compensate-for-missing-values-data-imputation-with-examples-6022d9ca0779). Gaps can cause problems in your modeling. Some models (for example ARIMA for time series) won't work at all if you have gaps that aren't handled. Looking at your use case, I think taking the last known value for a gap should work fine since a gap means your customer didn't churn on that day.


  The dates repeat for different users, should I create index out of a date but it won't be unique or should I create index out of customer_id?


Probably go for indexing the customer_id and creating features based on the date! Maybe there are some seasonality features interesting as well (for example how many people churned on the same day last year?)


  Q3: If I need to predict whether the user is going to churn by the end
  of the year for example or in the next 6 months, would that change how
  I group/arrange my date and model this?


You would have to change the size of the period you are predicting for. Changing features, grouping, etc. might be useful as well but that really depends on what features you are choosing aka are important and how your model is going to work.