I think this answer isn't more than an attempt to answer and my answer unfortunately don't include English literature sources (but I tried to add links to an Arabic dictionary to some of the key-words):
Definition: Of the two names of Mercy (ar-Rahman and ar-Raheem)
(MY OWN TRANSLATION TAKE IT CAREFULLY)
الرحمن/ الرحيم:
The names ar-rahman (the Entirely Merciful) and ar-raheem (the Especially Merciful). Both are related to ar-rahmah الرحمة the Mercy
الرقة والتعطف، والمرحمة مثله، ...، والرحمة المغفرة
Which in Arabic has the meanings: Gentleness and condescendence. Al-Marhamah المرحمة (clemency) is similar, an other synonym of ar-Rahmah is forgivness.
، وقوله تعالى في وصف القرآن {هدى ورحمة لقوم يؤمنون}؛ أي فصلناه هاديا، وذا رحمة،
Allah has described the Quran in (7:52)
"... as guidance and mercy to a people who believe."
this means we have detailed this book to make it a guidance (for the believers) and of (or containing) mercy (for them)
وقوله تعالى:{ورحمة للذين آمنوا منكم }؛ أي هو رحمة لأنه كان سبب إيمانهم ....
Allah has described his Messenger () in (9:61) saying
"...believes the believers and [is] a mercy to those who believe among you...."
This means he is a mercy because he was a reason for you to vecome believers.
والله الرحمن الرحيم، بنيت الصفة الأولى على فعلان، لأن معناه الكثرة، وذلك لأن رحمته وسعت كل شيء، وهو أرحم الراحمين، فأما الرحيم: فإنما ذكر بعد الرحمن لأن الرحمن مقصور على الله عز وجل، والرحيم قد يكون لغيره،... ورحيم فعيل بمعنى فاعل
Allah is ar-Rahman ar-Raheem. Ar-Rhaman in Arabic has a special linguistic structure which makes it refer to a lot/many/much of Mercy, as Allah's Mercy includes everything. And HE is the most Merciful of the Merciful. While Ar-Raheem was quoted after ar-Rahman because the later is exclusive for Allah, the Exalted , the Majestic, while ar-Raheem can apply to others (too). Raheem in the Arabic language has a structure, which make it refer to the doer.
Note that ar-Rahman is also a expresses exaggeration. And if we say Allah is ar-Raheem we mean He is the one who gives Mercy.
Now let’s take a look in a more theological definition
الرحمن هو العطوف على عباده بالإيجاد أولا .. وبالهداية إلى الإيمان وأسباب السعادة ثانيا .. والإسعاد في الآخرة ثالثا ، والإنعام بالنظر إلى وجهه الكريم رابعا .
الرحمن هو المنعم بما لا يتصور صدور جنسه من العباد ،
Ar-Rahaman is the one who is gentle towards his servants due to his creation firstly ... due to his guidance to belief and the reasons of felicity secondly... by bringing happiness in the hereafter thirdly, ... by the benefit to see His honorable face fourthly.
Ar-Rahman is the donor of all that which is unimaginable to be given or rise from his servants.
والرحيم هو المنعم بما يتصور صدور جنسه من العباد
While ar-Raheem is the donor of all that which is imaginable to come from his servants.
Defintion of Justice
Justice or al-'Adl العدل:
العدل ما قام في النفوس أنه مستقيم وهو ضد الجور ...،
'Adl means is what we (or our minds and conscience) understand as correct and it is the opposite of inequity
وفي أسماء الله سبحانه العدل: هو الذي لا يميل به الهوى فيجور في الحكم،
And among HIS beautiful names is al-'Adl: That is the One who doesn't decline due to fantasy or caprice and aggrieve in his verdict.
So we learn that Allah can't give a verdict as a result of whim!
Another definition says:
العدل من أسماء الله الحسنى ، هو المعتدل ، يضع كل شيء موضعه ، لينظر الإنسان إلى بدنه فإنه مركب من أجسام مختلفة، هي: العظم.. اللحم .. الجلد ..، ...، فلو عكس الترتيب وأظهر ما أبطن لبطل النظام ،
Al-'Adl is one of Allah's beautiful names, it also refers to the "correct" who puts everything in the right place, if the human looks at his body than he would findout that it is built from different materials and parts ...muscles, bones, skin ... if it was built up in a different order the whole system wouldn't work.
هو العدل الذي يعطى كل ذي حق حقه ، لا يصدر عنه إلا العدل ، فهو المنزه عن الظلم والجور في أحكامه وأفعاله ،
His justice is of the kind that appoints for everyone who has a right what is due to him (see for example in the hadith), only justice can be rendered by/from Him, as He is infallible and it is impossible to Him to be unjust or aggrieve in His verdict and actions.
Therefore Quran asks us to choose the midway and to apply justice in many verses:
...And thus we have made you a just community that you will be witnesses over the people and the Messenger will be a witness over you. ...(2:143)
...and when you judge between people to judge with justice. ...(4:58)
...And [they are] those who, when they spend, do so not excessively or sparingly but are ever, between that, [justly] moderate ...(25:67)
Relation between both
From above we know that Allah's mercy includes gentleness of Allah towards his servants and his guidance and many other donations he might give all of his servants or those who follow his guidance in the hereafter in special. While his justice can be summarized in two things putting everything in the right place and not abusing or aggrieve any of his creations. This already seems to show that there's no intersection rather than a clear cut between both.
Here an attempt to explain how they don't contradict each other which is to some extent a copy-paste from my source below with some comments of mine.
So whoever does an atom's weight of good will see it,(7) And whoever does an atom's weight of evil will see it. (99:7-8)
So on Judgementday anything will be put on the weighing scale.
And there is no creature on earth but that upon Allah is its provision, and He knows its place of dwelling and place of storage. All is in a clear register. (11:6)
Allah's mercy covers all kinds of His creation and all "information" about them is in a record.
Say to those who have disbelieved [that] if they cease, what has previously occurred will be forgiven for them. But if they return [to hostility] - then the precedent of the former [rebellious] peoples has already taken place. (8:38)
Allah's mercy includes the disbelievers (in this life) and they would be rewarded by his forgiveness in case of conversion. Else they would be punished according their committed sins. So they may learn about Islam or Allah and accept Him as their creator or reject Him (because of stubbornness for example) in this life.
We have enjoined on man kindness to his parents: In pain did his mother bear him, and in pain did she give him birth. The carrying of the (child) to his weaning is (a period of) thirty months. At length, when he reaches the age of full strength and attains forty years, he says, 'O my Lord! Grant me that I may be grateful for Thy favour which Thou has bestowed upon me, and upon both my parents, and that I may work righteousness such as Thou mayest approve; and be gracious to me in my issue. Truly have I turned to Thee and truly do I bow (to Thee) in Islam.' (46:15)
"Lord of the heavens and the earth and whatever is between them, the Exalted in Might, the Perpetual Forgiver." (38:66)
Those who avoid the major sins and immoralities, only [committing] slight ones. Indeed, your Lord is vast in forgiveness. He was most knowing of you when He produced you from the earth and when you were fetuses in the wombs of your mothers. So do not claim yourselves to be pure; He is most knowing of who fears Him. (53:32)
Allah knows all about us he even knows our weakness therefore it is part of His mercy to forgive once we repent or learn -what is right and or wrong- from our mistakes.
In these Noble Verses we see the following:
1- Allah Almighty records everything that His creations do, and will bring it to their attention on the Day of Judgement.
2- Allah Almighty recognizes that as the person grows up, he/she learns. His example of the age of 40 in Noble Verse 46:15 clearly proves this.
3- Allah Almighty's Mercy and Forgiveness are vast and He always Forgives His creation.
With these points established from the Noble Quran, we can now clearly see how Allah Almighty's Mercy and Justice do not contradict each other’s in anyway. (source)
I'd like to add some verses showing his justice a bit more on judgement day such as: (17:13-14) and (18:48-49).
Which show that on judgement day we will be presented our records and would therefore accept the verdict, as it is based on clear evidences.
Sources for the definitions: achaari and alhekma.