Having said "Lo Sateh Mishpat", why does the Pasuk add "Lo Sakir Panim"?
Rashi: It is forbidden to favor either of the litigants even whilst they are presenting their cases 1 - to speak softly to one of them and harshly to the other, or to allow one of them to sit whilst the other one stands. 2
Having said "Lo Sateh Mishpat", why does the Torah need to say "ve'Lo Sikach Shochad"?
Rashi #1: To teach us that a Dayan may not accept bribes even to judge correctly. 1
Rashi #2: Because once a Dayan accepts a bribe from one of the litigants, it is impossible for him not to turn his heart against the other litigant.
See also Sifsei Chachamim.
What has "Viyesalef Divrei Tzadikim" got to do with Beis-Din?
Rashi, Targum Onkelos and Targum Yonasan: It means that bribery twists the righteous words (in the mouths of the Dayanim during litigation - Targum Yonasan).
What does the Torah mean when it writes "Tzedek ... Tirdof"?
Ramban #2 (citing the Ibn Ezra #1): 1. That the litigants should always tell the truth, irrespective of whether they will gain or lose by it; 2. That they should continue to do so time after time.
Ramban #3 (citing the Ibn Ezra #2): the Torah writes it twice for the sake of emphasis.
Seforno: When appointing the Dayanim, one should pick those who are able to judge righteously, over and above the other attributes that a Dayan ought to posses - such as good health and wealth. 3