What is the proper way to calculate bootstrap regression coefficients CI after MI? I mean is using a pooled bootstrap SE over imputed data sets with a pooled estimate of bootstrap regression coefficient across MI data sets better or using list of bootstrap SE and regression coefficients estimates for each MI data sets? The former approach gives one CI while later gives a list of CI for each MI data set.
I suggest looking at this recently published paper by Schomaker and Heumann, 'Bootstrap inference when using multiple imputation'. The published version in Statistics in Medicine is here. A freely accessible pre-print version on arXiv is here. Some comments I made on the arXiv version before it was published and discussion with the authors can be found here.
A somewhat different approach using bootstrapping with multiple imputation was described by von Hippel in this arXiv paper.
I recently published a paper which investigates a number of different ways of combining bootstrap and multiple imputation, and in particular how the order in which you do things affects validity when the imputation and analysis models are so called 'uncongenial' or misspecified.