Why does the Torah see fit to begin with the words "Hi Mutzeis" (she was being taken out, i.e. to be burned)?
Seforno: To teach us that Tamar was as tough as a lion, in that, even in that eleventh hour, when all seemed lost, she did not lose her composure, and fended for herself with determination. 1
Riva, citing R. Chananel: It says Mutzeis without an Aleph. 2 This hints to read it like an expression of kindling, [to teach that it is better to be burned than to humiliate someone].
Why did Tamar not confront Yehudah directly and announce that it was from him that she was pregnant?
According to Targum Yonasan, she was initially unable to find them, until, after praying fervently to Hashem promising Him that, if He would help her to find the three 'witnesses', she would supply Him with three holy men, who would sanctify His Name by entering a burning furnace in the valley of Dura. At which point Hashem signaled to Micha'el to show her where they were.
As the Gemara says in Sotah, 10b 'It is preferable to allow oneself to be cast into a burning furnace than to shame one's friend in public' (Rashi). Moshav Zekenim (24) - Misas Sereifah is through pouring molten lead down the throat. Beis Din does not put the person in a furnace! This requires investigation.
What is the significance of the words "Haker Na"?
And why did she add the word "Na"?
Targum Yonasan: She was pleading with Hashem to put in Yehudah's heart to acknowledge that the three articles were indeed his.
What was Yehudah's initial reaction to Tamar's pleas?
Targum Yonasan: Yehudah figured that it is better to be put to shame in this passing world, than in front of his fathers in the everlasting World to Come, and that it is preferable to be burned in a physical fire (with reference to becoming embarrassed) than to be burned in the consuming fire of the World to Come. 1 And he acknowledged that Hashem was punishing him Midah ke'Neged Midah. 2
Ohr Yechezkel Darchei ha'Avodah p.222: This was due to his Midah of Emes, which includes recognizing the eternity of the world to come, as opposed to the folly of this [fleeting] world.
Refer to 38:25:3:1. None of this is even hinted at in the Pasuk. [Sefer ha'Itim (end of #175) writes that Chazanim, who translated during Keri'as ha'Torah, used to add Midrashim to Targum Eretz Yisrael, and they became incorporated into the Targum. Subsequently, the compilation mistakenly became called Targum Yonasan; Yonasan himself did not translate the Torah, see Megilah 3a]
רש"י: ואל תאבד שלש נפשות: מנין ידעה שיש לה תאומים?
גור אריה: קבלה הייתה בידם שהמלכות תצא מיהודה, ומכיוון שמלכות יהודה משולה לירח, צריך שיוולדו תאומים ליהודה שהרי אין ירח בלי שמש.
רש"י: נח לו לאדם שיפילוהו לכבשן האש וכו': למה כך הדבר?
מהר"ל (חידושי אגדות ח"ב, עמ' מג, סוטה דף י: ד"ה נח): כבשן האש הוא כליון גמור ואעפ"כ הלבנת פנים היא הפסד גדול יותר, כי הכבשן מפסיד רק את הגוף הטבעי ואילו המלבין פני חברו מפסיד ומבטל את הצלם האלוקי של חברו.