By a gentile who is under your control - בעובד כוכבים שישנו תחת ידך
We can establish the פסוק of וחשב עם קונהו to be discussing a נכרי שאינו תח"י and nevertheless the תורה teaches us that we should advise him of the Jewish law, and he should not be deceived. In this case where the נכרי שתח"י trusted that he would be compensated fairly it is forbidden to cheat him.
Since it takes out a Jewish daughter - שכן מוציא בבת ישראל
It is preferable to bring proof from something which is written explicitly in the תורה (even though it is not that similar), than to bring proof from something which is not written explicitly in the תורה (even though it is more similar).
Let him say to him in the presence of two; Go! - לימא ליה באפי תרי זיל
An ע"ע can be freed through הפקר. The question of למה לי שטרא is that the very same written words can be said orally (since we assumed that there is only a monetary obligation owed to the master). The conclusion is that גופו קנוי requires a שטר to nullify the קנין הגוף.
And the master who forgave his deduction - והרב שמחל על גרעונו
There is no need for a קנין if the אדון was מוחל the ע"ע to make the שחרור effective (if we assume אין גופו קנוי), because the עבד is מוחזק בגופו; however by other debts perhaps מחילה requires a קנין as well.