Did Yosef have special servants who were doctors?
Malbim: Yes. This shows his grandeur!
Why is embalming called 'Chanatah'?
Da'as Zekenim: After they put bitter ingredients to remove the waste from the innards, lest it rot, they put in sweet ingredients for a pleasant smell. This is like 'Chanatah' of unripe figs; they become sweet.
Malbim (to 50:3): They believed that it resembles development (refer to 50:3:151:1) - as in "ha'Te'enah Chantah Fageha" (Shir ha'Shirim 2:13).
Why did Yosef embalm his father?
Ohr ha'Chayim #1: He did so for his father's honor. This is done for esteemed people, and all the more so for the esteemed of the kingdom.
Ohr ha'Chayim #2: This was lest people err and say that he did not die, 1 or that he died and did not rot, and they will make him a god, 2 and not allow burying him in Eretz Yisrael. Surely even without embalming, he would not have rotted, like we find regarding R. Elazar ben R. Shimon (Bava Metzia 84b).
Ohr ha'Chayim #3: When Bnei Yisrael stood on Har Sinai, their filth departed (Shabbos 146a). From then and onwards, Tzadikim of Yisrael did not rot after death. Even waste in the innards [does not rot; it] dries and turns to earth, for the filth of the body makes food in the innards rot. This did not apply before [Matan Torah]. Yaakov's body itself would not rot, but after a while, the waste in his innards would rot. Therefore, Yosef embalmed him.
Ohr ha'Chayim #4: Perhaps Yosef did not know [that Tzadikim do not rot].
Malbim: The Mes needs burial. The body must decay in the earth. The Divine Neshamah separates from the body immediately after death and returns to its source, but the Nefesh (Ru'ach) does not. In life, via investigation and deed one can purify it, i.e. cut off its attachment to the physical. Not everyone does this fully, for the investigation is via senses, and physical motives are mixed with the deeds. Only after the flesh returns to the earth, the Ru'ach returns to its spirituality. Tum'as Mes is due to these parts tied in the Mes' flesh until it decays. A Kadosh like Yaakov totally has purified his Ru'ach, so there is no Tum'ah; the body need not rot. The Egyptians embalmed, for there was no reason to bury them; their Nefesh and Ru'ach became physical, and there is no Tum'ah, just like an animal! 3 Yosef embalmed Yaakov's body to remain like a Tahor garment that his awesome Nefesh wore. He is called Yisrael in this Pasuk, for he was at the level of Yisrael.
What is the problem if they say so? Perhaps when Yosef will ask to go to bury him, they will say that it is a false pretext. It would not be honorable to open his coffin to prove it, or then there is great concern lest they deify him, when they see that he did not rot! (PF)
Would people consider him to be a god after he died?
A dead animal has Tum'as Nevelah, but not the severity of Tum'as Mes. Also Egyptians buried their dead (Bamidbar 33:4)! Perhaps this was only to avoid stench and nuisance of their decay; only their leaders were embalmed. Or, perhaps only men knew how to embalm, and they all died in Yam Suf (see Rashi to Yevamos 76b DH Hanach - PF).
Was it proper for Yosef to embalm his father?
Refer to 50:26:152:1 and the note there.
Ohr ha'Chayim gives four explanations why Yosef did so (refer to 50:2:152:1, and on). According to his last Perush, Yosef was unaware that there was no need to do so; the other Perushim imply that he did properly. (PF)
Malbim implies that it was proper; refer to 50:2:152:5. (PF)
Ha'amek Davar: No. One opinion in Bereishis Rabah 100:3 says that this is why Yosef died before his brothers! Hashem said - I cannot guard my Tzadikim?! Did I not say "Al Tir'i 1 Tola'as Yaakov" (Yeshayah 41:4) - (interpreted as, worms will not see Yaakov; he will not decay! 2 ) He is called Yisrael, to teach why Hashem was angry that Nochri doctors embalmed one who was at the level and Kedushah of Yisrael.
In the verse, there is a Yud after the Tov; which is an expression of 'fear.' The Derashah reads it without a Yud, an expression of 'seeing.' (PF)
Harchev Davar: Did Yosef think that worms ate Avraham and Yitzchak, who were not embalmed?! Perhaps Yaakov commanded not to be buried even temporarily in Egypt lest they need to take some surrounding earth. Yosef understood that he may leave him for a while without burial. Yosef did so for Yaakov's honor. The crying and mourning lasted a long time; Yosef feared lest he decay on the bier, unlike Avraham and Yitzchak, who were buried immediately. Also the Segulah of Me'aras ha'Machpelah would help [avoid decay]. Hashem was angry, for Yosef should have realized that a Tzadik is guarded in any case.