3) 'Search' lets you keep the two drawn cards? What keeps the 'no hand limit' policy from being overpowered? I'm sure there is some reason, just wish to understand better.
Search is an action. One of 3 you can do... So in general you do less stuff in your turn. you can of course 3x search, but beside 6 cards for future use it will give you nothing. You will not do anything on the table. And before you may be able to use those cards - opponents can grab some points, close portals, complete set of 3/5... With 2 players maybe it will work to do few searches, but with 4/5 - it will slow you down tremendously and in final score will not help you at all.
Beside you always draw to 3 cards in hand at the end of your turn. If you have more cards you draw nothing. That equals to you doing less actions in your turn than opponent who did all 3 "table" actions.