Why did God send prophets, & how did He choose them?

The corrupted Holy Book doesn't give us a clear & comprehensive idea about (prophecy & prophets). There is no standard why God chose them, & when you ask the people of the Book they answered you that, "It's God's will"!

Ok, but God isn't moody, or unfair, His will is always accompanied by His Wisdom. What makes a person get that honor i.e. being a prophet of God? Why did God choose Moses not Jushua for example to be His prophet? Why did He choose some sinners & liars to be His prophets?

"It's God's will"!

Prophets in the Holy book are sinners just like ordinary people, moreover, their crimes aren't simple, Moses killed a man deliberately (Exodus 2: 12), they may disobey God (Jonah), did things which contradict monotheism (e.g. Jacob set up a pillar stone; Genesis 35: 14 & Aaron made a golden calf;Exodus 32), lied to others (as Jacob lied to his father; Genesis 27: 24, also a prophet lied to another prophet; 1Kings 13: 18), committed adultery (e.g. David & Lot), curse others for small mistakes or misunderstanding (as Noah cursed Canaan; Genesis 9: 25 & Jeremiah in his chapter 20: 14,15 & 20: 18).

You can't consider them good examples & standards for perfection; nor can you follow them in all their deeds.

How can I take my religion from a liar, or how can I follow a misleader, or how can I trust wrong-doers who contradicted themselves & be the first who break the laws??

In fact, those who corrupted the Holy Book & changed it smudged prophets' reputation & attributed horrible sins to them.

In the contrast, The Holy Quran told us that God The All Wisdom chose those who were the best among their people, they're all sinless & faultless, no one of them had ever lied to others or misled them or committed sins, yes they may forget as Adam PBUH did "115 And We had already taken a promise from Adam before, but he forgot; and We found not in him determination" Surat Taha

Or they may mis-choose the best, as prophet Mohammad PBUH did, once he was busy in persuading some big chiefs of Mecca to accept Islam, meanwhile a blind man came to ask about Islam & interrupted him, but the holy prophet PBUH disliked his interruption and ignored him, so God blamed him: " He frowned and turned away his face because there came up to him the blind man. And what would make you know that he might reform, or heed the admonition, and admonishing might profit him? As for him who is indifferent, to him you attend, though you would not be responsible if he did not reform. And the one 'who comes to you running, of his own will, and fears, from him you turn away. By no means! This is but an admonition." Surat Abasa 1-11

Or maybe they act hastly as when David PBUH judged between the litigants, he judged immediately without listening to the opponent), but he guessed that God had tried him "and he sought Forgiveness of his Lord, and he fell down prostrate and turned (to Allah) in repentance" 24 Surat Saad

We can't call those sins 'disobedience or offence'. They are kind of 'mistakes'.

What is their main point & job in the Holy Book?

Their main point is the relationship between God & His chosen people 'Israel' & the relationship between Israel & other nations, so they seem to be political analysts (e.g. Isaiah), consultants (Chronicles II 25: 7,8), commanders (e.g. Moses & David), reformers (most of them), also one of their important tasks is seeing prophesies & visions (e.g. Balaam, Micah, & Daniel).

Ok, but anybody can take on political affairs. Living in peace & prosperity aren't everything, there is something more important ignored by the Holy Book, i.e. the Hereafter.

Prophets in the Holy Book give glad tidings to obedients that they shall have the peace and prosperity of the kingdom (e.g. but he that putteth his trust in me shall possess the land, and shall inherit my holy mountain) Isaiah 57: 13

And they warned disobiedients from the pestilence, famine, & the sword (e.g. Ezekiel 5: 12)

They didn't focus on the Hereafter, the Paradise, the Hell, & God's satisfaction, few of them mentioned that, as Amos, Obadiah, & Zephaniah, but others ignored it! E.g. when David lamented the passing of Saul and Jonathan, he didn't mention anything about resurrection, Hereafter, God's mercy, instead; he said "27 How are the mighty fallen, and the weapons of war perished!" 2 Samuel 1

The Holy Quran gives us a clearer idea about prophecy; it tells us that their main job is to lead people to worship God alone
& to believe in the Hereafter,
they are bringers of good tidings "63 Those who believed and were fearing Allah 64 For them are good tidings in the worldly life and in the Hereafter" Surat Yunus,
and warners e.g. Noah said, "O my people, worship Allah ; you have no deity other than Him. Indeed, I fear for you the punishment of a tremendous Day." ( Al-A'raf 59).

Quran focuses on their debate with the disbelievers, their main point isn't political affairs, but leading people to the belief in their Lord, in addition to calling for reforms.
" 84 And to Madyan [We sent] their brother Shu'ayb. He said, "O my people, worship Allah; you have no deity other than Him. And do not decrease from the measure and the scale. Indeed, I see you in prosperity, but indeed, I fear for you the punishment of an all-encompassing Day.85 And O my people, give full measure and weight in justice and do not deprive the people of their due and do not commit abuse on the earth, spreading corruption" Surat Hud

Also they led us to the legislations that organize our lives. (Al-An'am 44-48)

When God sends a teacher for you, & orders you to follow him, he must be a good example for you. Therefore, God completed their virtues, preserved them from sins & made them perfect examples & standards for good manners & perfect personality.

In the Holy Quran God praised prophets & ordered us to follow them,
"Say, [O Muhammad], "Praise be to Allah , and peace upon His servants whom He has chosen" 59 Al-Naml
they are our reference not only in worship & obedience, but also in our lifestyle.
"Those are the ones whom Allah has guided, so from their guidance take an example" 90 Al-An'am

Adam taught us to repent & be aware of Satan, Noah taught us to continue to the end without desperation & to fulfill our work as he fulfilled the ark, Ibrahim taught us how to debate disbelievers using proofs & logic, David taught us to depend on ourselves, Joseph was a perfect example of chastity, also he gave us basics in economy & interpreting the dreams, Moses taught us to face danger courageously, Solomon taught us to be thankful to God & to impress disbelievers with our civilization, Job taught us to be patient, Jesus taught us to be kind with our mothers & to be supporters of Allah, Mohammad taught us to be positive, clean, honest, merciful, humane, generous, to take care with other people, animals, & plants, he taught us to fear God & follow his prophets peace be upon all of them.

Wherever we follow them we'll be well-guided and successful.