Welcome to the forum:)
Regarding your questions on non-beleivers and what will happen to them on the Day of Judgement.
If the person received the clear message of Islam and rejected it then there will be evidence against them. Allah (SWT) forgives all sins except the sin of setting up "partners" with him (ie. rejecting/disbelieving/making false gods), and inevitably the one who rejects Islam after having understood and hearing the message of Islam then there abode will be hell.
However if a person lives there life without ever hearing the clear message of Islam then they will go through a test on the Day of Judgement to see whether they would have accepted Islam or not.
So you see everyone will have the opportunity too accept Islam, for some it will be in this life and others will have that chance on the Day Of Judgement.
No injustice will be done to anyone, if they are a disbeliever they will be paid back there good in this life and if they are a believer there reward will be in this life and the hereafter. As Allah (SWT) reitirates throughout the Qu'ran:
"The Word changes not before Me, and I do not the least injustice to My Servants." (50:29)
"If anyone does a righteous deed it ensures to the benefit of his/her own soul; if he/she does evil, it works against it [his/her own soul]" (45:15).
“And your Lord is not unjust to His slaves.” (41:46)
“Indeed Allah is not unjust to mankind but it is mankind who are unjust to themselves.” (10:44)
"Verily God will not deal unjustly with man in aught; it is man that wrongs his own soul. One day He will gather them together; [it will be] as if they had tarried but an hour of a day; they will recognize each other; assuredly those will be lost who denied the meeting with God and refused to receive true guidance"
Then anyone who has done an atom's weight of good, shall see it And anyone who has done an atom's weight of evil, shall see it (99:7-8).
Regarding the second part of your question. Allah (SWT) guides those who are sincere in there search for guidance. As Allah (SWT) says in the Qu'ran:
"..As for those who strive hard in Us (Our cause), We will surely guide them to Our paths. And verily, Allah is with those who excel in good." Al-Ankaboot(29):69
"The same religion has He established for you as that which He enjoined on Noah - the which We have sent by inspiration to thee - and that which We enjoined on Abraham, Moses, and Jesus: Namely, that ye should remain steadfast in religion, and make no divisions therein: to those who worship other things than Allah, hard is the (way) to which thou callest them. Allah chooses to Himself those whom He pleases, and guides to Himself those who turn (to Him). (42:13)"
Read up on the true message of Islam from reliable sources and continue to ask God for guidance inshallah (God Willing) he will open up your heart and understanding to Islam, be patient and don't lose heart in your quest. Any questions you have feel free to post.