Interpretation of 5:43
The verse in context is as follows:
فإن جاءوك فاحكم بينهم أو أعرض عنهم ... وكيف يحكمونك وعندهم التوراة فيها حكم الله ثم يتولون من بعد ذلك وما أولئك بالمؤمنين ... وأن احكم بينهم بما أنزل الله ولا تتبع أهواءهم واحذرهم أن يفتنوك عن بعض ما أنزل الله إليك ... أفحكم الجاهلية يبغون ومن أحسن من الله حكما لقوم يوقنون
So if they (the Jews) come to you, [O Muhammad], judge between them or turn away from them ...
But how is it that they come to you for judgement while they have the Torah, in which is the judgement of Allah? Then they turn away, [even] after that; but those are not [in fact] believers.
And judge, [O Muhammad], between them by what Allah has revealed and do not follow their inclinations and beware of them, lest they tempt you away from some of what Allah has revealed to you ...
Then is it the judgement of [the time of] ignorance they desire? But who is better than Allah in judgement for a people who are certain [in faith].
— Quran 5:42-50
This verse was revealed when the Jews of Medinah came to the Prophet ﷺ seeking a judgement on the case of a murder and\or adultery. They wanted him to give a verdict against the law of the Torah (i.e. they desired inequality in retaliation for murder and public humiliation rather than stoning for adultery).
This verse condemned them for following their desires and wanting to impose a man-made law, and demanded that the law of the Torah be enforced. And the law for murder and adultery is the same in Islam as in the Torah.
Does 5:43 mean that Jews should be judged by the Torah?
And how does the Torah contain the law of Allah if it is distorted?
They must judge and be judged (by Muslim rulers) according to the Quran and what is confirmed from the Torah in the Quran and Sunnah. The Torah and the Gospel are not fully corrupted, and there is difference among the madhabs on whether the text is corrupted or only its interpretation. Anything which is confirmed in the Quran is obviously correct.
Interpretation of 5:47
In context the verse is as follows:
وقفينا على آثارهم بعيسى ابن مريم مصدقا لما بين يديه من التوراة وآتيناه الإنجيل فيه هدى ونور ومصدقا لما بين يديه من التوراة وهدى وموعظة للمتقين
وليحكم أهل الإنجيل بما أنزل الله فيه ومن لم يحكم بما أنزل الله فأولئك هم الفاسقون
And We sent, following in their footsteps, Jesus, the son of Mary, confirming that which came before him in the Torah; and We gave him the Gospel, in which was guidance and light and confirming that which preceded it of the Torah as guidance and instruction for the righteous.
And let the People of the Gospel judge by what Allah has revealed therein. And whoever does not judge by what Allah has revealed - then it is those who are the defiantly disobedient.
— Quran 5:46-47
This can mean the following:
Allah sent the Injeel so that the followers of Jesus could judge by it. This was applicable from the time of the revelation of the Injeel until the revelation of the Quran and is now no longer applicable.
It is still applicable in the sense that the Injeel commanded the Christians to follow Muhammad ﷺ (see 7:157 , 61:6, 48:29 and 5:12 etc.) , hence what this means is that the Christians must accept Islam following the command from the Injeel.
It is still applicable in the sense that they must follow those laws of the Injeel which are affirmed in the Quran and not abrogated.
It does not mean that following the present Gospel is acceptable for the present Christians. Rather, there are numerous texts within the Quran and ahadith that firmly establish that nothing is accepted from them by Allah other than conversion to Islam.
So should Christians judge by the Gospel, or not?
And how can they use the gospel for judgment if it is wrong?
They should follow the Gospel in the sense that they should accept Islam. And they should judge and be judged (by Muslim rulers) according to the Quran and that part of the Gospel which is confirmed in Islamic sources.
Interpretation of 5:48
In context, the verse is as follows:
إنا أنزلنا التوراة فيها هدى ونور يحكم بها النبيون الذين أسلموا للذين هادوا ... وقفينا على آثارهم بعيسى ابن مريم مصدقا لما بين يديه من التوراة وآتيناه الإنجيل ... وأنزلنا إليك الكتاب بالحق مصدقا لما بين يديه من الكتاب ومهيمنا عليه ... لكل جعلنا منكم شرعة ومنهاجا ولو شاء الله لجعلكم أمة واحدة ولكن ليبلوكم في ما آتاكم
Indeed, We sent down the Torah, in which was guidance and light. The prophets who submitted [to Allah] judged by it for the Jews ... And We sent, following in their footsteps, Jesus, the son of Mary, confirming that which came before him in the Torah; and We gave him the Gospel ... And We have revealed to you, [O Muhammad], the Book in truth, confirming that which preceded it of the Scripture and as a criterion over it ...
To each of you We prescribed a law and a method. Had Allah willed, He would have made you one nation [united in religion], but [He intended] to test you in what He has given you; so race to [all that is] good. To Allah is your return all together, and He will [then] inform you concerning that over which you used to differ.
— Quran 5:44-48
This verse simply says that each successive Ummah was given its own Shariah. It is addressing people who were alive at different times. It does not allow the Jews and Christians from the present time to follow the old Shariah rather than the Quran.
5:48 clearly says that Allah has established separate way and
law for Jews and Christians ... So Jews and Christians are
following the path set by Allah
It does not say that at all. It is not talking about the present Jews and Christians, rather it is about the Moses عليه السلام and his followers till before Jesus عليه السلام and about Jesus عليه السلام and his followers till before Muhammad ﷺ. As for after receiving the message of Muhammad ﷺ: nothing is accepted by Allah other than Islam.
يا أيها الذين أوتوا الكتاب آمنوا بما نزلنا مصدقا لما معكم من قبل أن نطمس وجوها فنردها على أدبارها أو نلعنهم كما لعنا أصحاب السبت وكان أمر الله مفعولا
O you who were given the Scripture, believe in what We have sent down [to Muhammad], confirming that which is with you, before We obliterate faces and turn them toward their backs or curse them as We cursed the sabbath-breakers. And ever is the decree of Allah accomplished.
— Quran 4:47
And you can further refer to my answers here and here for many other texts which say the same.