panel data - within-group estimate - individual fixed effects retrieved

#1
Hi TalkStats people!

I am analyzing panel data.

First, I have to decide whether to use a random or fixed effect estimator.
The Hausman test suggests to use the fixed effect estimator (also named within group estimator). Thus, this is what I am using.
However, this method eliminates the individual fixed effects, that is, the Ui's, which is what I am more interested about. Hence, I proceed with a second step.

Second, I run a between-group estimator where I regress the predicted individual effects, that is the Ui's predicted from the first step, against a list of time invariant fixed effects I am interested in

My questions are three:

1. I was taught this two-step method in a graduate summer course, and no book I have read mentions it, do you know what is its name?
2. Is it meaningful to cluster the standard errors on individuals in the first step? (i.e. within-group estimator)
3. Is it meaningful to use a variable both as a covariate and as cluster for the standard error in the same model?

Thank you for your time!