The statement below is false.
The statement above is correct.
PDS fire happens during the move step of an activation. It does not happen during a retreat/withdrawal step of a combat round. Thus, your PDS cannot fire at retreating/withdrawaling ships.
This is correct. More precisely, it's step three of both the tactical action and the transfer action, see pages 10 and 12 respectively.
Also in the same game in the end - two of us could claim 2vp after that round. I had Tech card and he had War II card. I assumed he won 'cause his initiative was lower than mine but then it hit me - shouldn't we pick the cards for that round and THEN claim objective we can which means I would be the winner as I was the speaker and naturally would pick the lowest initiative card then.
Nope. You claim strategy cards in the strategy phase of a round, which is the first phase. You claim victory points as the first step of the status phase of a round, which is the last phase, and the seventh step of the status phase is returning the strategy cards. Have a look at page 14 of the base game rulebook for all the steps of a staus phase, their order is quite clearly defined.