The answers are yes, yes and yes.
On a loop pile wool carpet agitation with a stiff pile brush could sometimes cause issues. It's just something you need to gauge on every job.
Where possible agitation after prespraying will improve the results of extraction immensely. It enables the prespray to be distributed around all the fibres properly so it can do its job properly. It also opens the carpet up, lifting the pile and helping to restore matted areas.
Grooming the carpet after extraction is also good practice. Its a bit like combing your hair after washing it. It removes all the lines left by the extraction process. The carpet tends to want to lay in a particular direction, that's the best way to brush it for the best looking results.
When agitating the carpet with a pile brush I like to brush backwards and forwards quickly across the carpet in lines East to west then again north to south. Like you would if you were scrubbing something with a broom. Of course every situation is different so its important to think about what you are trying to achieve rather than just following steps blindly. Some carpets I would agitate with a rotary machine some with a crb, some with a brush. Some not at all. It will be fine to use a brush on 90% of carpets though.