Star Realms: scrapping a card in your hand
Your interpretation is incorrect. You cannot use the actions of a card that you scrap.

The rules state that during the main phase, players may perform the following actions in any order (among other actions):


  
  Play cards from your hand
  Use the Primary Abilities of in-play Bases
  


So the answer to your second question is that you can use the Brain World in whatever order you want during the main phase.

The rules go on to state 


  When a Ship is played, its Primary Ability triggers immediately.


That is to say, you must decide to play a card from your hand in order to gain the benefit of the primary ability. 

Under "Playing Cards" they say that to play a card, 


  place it "in-play" (face-up on the table in front of you).


Later, in the Discard Phase,


  Put all in play Ships into your Discard Pile.
  Put any cards left in your hand into your Discard Pile. 


These quotes together make it clear that an "in-play" card is not in your hand, nor your discard pile. This demonstrates that using a Primary Ability and being scrapped by the Brain World are mutually exclusive (at least without some special ability to move cards from in-play to your hand or discard pile).