Assalaamu alaykum, Shaykh.
1) What are the pillars, obligatory parts, and Sunnan of the funeral prayer according to the Hanbali School of Thought?
2) In the Hanbali School of Thought, is it obligatory to recite Surah Al-Fatihah, or can a person recite the opening supplication instead?
3) If a person missed an obligation in the funeral prayer, what should he do given that he cannot do the prostration of forgetfulness?

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger. 

The Hanbali scholars described the funeral prayer, mentioning its obligations and Sunan (plural of Sunnah, i.e. recommended acts).

(The Hanbali book) ‘Umdat At-Taalib reads:

“The funeral prayer is performed as follows: the imaam stands in line with the chest of the (deceased) man and in line with the waist of the (deceased) woman, as an act of the Sunnah. He makes Takbeer (saying Allahu Akbar [Allah is The Greatest]) four times; after the first Takbeer, he recites the Faatihah after Ta'awwuth (seeking refuge with Allah from Satan). After the second Takbeer, he invokes blessings upon the Prophet, sallallahu ‘alayhi wa sallam, like in the final Tashahhud of the prayer. After the third Takbeer, he supplicates Allah for the deceased, saying, 'Allahumma ighfir lihayyina wa mayyitina wa shaahidina wa ghaa’ibina wa sagheerina wa kabeerina wa thakarina wa unthaana, innaka ta‘lamu mutaqalabana wa mathwaana, wa anta ‘ala kulli shay'in qadeer. Allahumma man ahyaytahu minna fa ahyihi ‘ala al-Islam was-sunnah, wa man tawaffaytahu minna fa tawaffahu ‘alayhima. Allahumma ighfir lahu warhamhu, wa ‘aafihi w-a‘fu ‘anhu, wa akrim nuzulahu wa wassi‘ mudkhalahu, waghsilhu bil-maa’i wath-thalji wal-barad, wa naqqihi min al-khataaya kama yunaqqa ath-thawbu al-abyadhu min ad-danas, wa abdilhu daaran khayran min daarihi wa ahlan khayran min ahlihi. Allahumma adkhilhu al-jannah wa a‘ithhu min ‘athaab al-qabri wa min ‘athaab an-naar, wa afsih lahu fi qabrihi, wa nawwir lahu fihi.' (Which means: O Allah, forgive those of us who are living and those who are dead, those of us who are present and those who are absent, those of us who are young and those who are old, those of us who are males and those who are females. O Allah, You know our changeable conditions in this life and our final destination, and You are over all things Able. O Allah, whoever of us You keep alive, keep him alive upon Islam and the Sunnah, and whoever of us You cause to die, cause him to die upon them. O Allah, forgive him and have mercy on him, keep him safe and pardon him, honor the place where he settles, and make his entrance wide; wash him with water and snow and hail, and cleanse him of sin as a white garment is cleansed of dirt. O Allah, give him a house better than his house and a family better than his family. O Allah, admit him to Paradise and protect him from the torment of the grave and the torment of Hellfire; and make his grave spacious and fill it with light.) If the deceased is a female, then all the pronouns should be feminine. If the deceased is a child, then instead of seeking forgiveness for him, the people should recite this supplication: 'Allahumma ij‘alhu thukhran li-waalidayhi wa faratan wa shafee‘an mujaaban, allahumma thaqqil bihi mawaazeenahuma, wa a‘thim bihi ujoorahumaa, wa alhiqhu bisaalih salaf al-mu’mineen, wa-j‘alhu fee kafaalati Ibraaheem, wa qihi bi-rahmatika ‘athaab al-jaheem.' (O Allah, make him a preceding reward, a stored treasure for his parents, and an answered intercessor. O Allah, by virtue of his death, make his parents scales heavy and magnify their rewards. Unite him with the righteous believers, place him under the care of Prophet Ibraaheem (Abraham), and protect him by Your mercy from the torment of Hellfire.) He pauses for a little while after the fourth Takbeer and then recites one Tasleem (saying 'Assalaamu alaykum') to the right. He raises his hands with each Takbeer. (He then underlined the obligatory parts of this description, saying:) standing up straight, the four Takbeers, reciting the Faatihah, invoking blessings upon the Prophet, sallallahu ‘alayhi wa sallam, supplicating for the deceased, and the Tasleem.”

Thus, it becomes known from the last part that, apart from the obligatory parts, if the person did something differently, his funeral prayer is still valid. For example, the mentioned supplication, making Tasleem to the right, raising the hands in Takbeer, the imaam standing at the chest of a man and at the waist of a woman, and other Sunnah acts that do not invalidate the prayer if they are left out. There is no prostration of forgetfulness in the funeral prayer. If the person misses any of the obligatory parts of the funeral prayer, he makes up for it in kind. According to the Hanbalis, it is not prescribed to recite the opening supplication in the funeral prayer because it is meant to be a short prayer. Reciting the Faatihah is obligatory, as has been mentioned, so the funeral prayer is invalid without it.

Allah knows best.