Thank you for your question. Your question actually includes multiple questions. I will try to answer all sequentially, but forgive me if I miss any.
Just so that it is easy for any reader, I will quote parts of your question followed by my answers.
The Qur'an testifies to the truth and authenticity of all the previous Scriptures
...Yes this is TRUE
And claims that all these heavenly books are a guidance, blessing and light
to all the people of the world.
TRUE that the Quran says these books were "guidance, blessing and light"
BUT FALSE that it is for "all the people of the world" or for this day and age. I have not seen any verse that says these.
All are to believe in these Scriptures.
FALSE, we do believe that the Gospel and Torah was revealed from Allah but this does not validate its current day authenticity or require us to believe in its present day content entirely.
They (Jews and Christians) were exhorted to hold fast to them
PARTLY TRUE, as per the verses you quoted (5:66,68) the Jews and Christians were exhorted to hold fast to the Torah, Bible AND THE QURAN. Read entire verses below...
"And if only they upheld [the law of] the Torah, the Gospel, and what has been revealed to them from their Lord,.." (Quran 5:66)
Say, "O People of the Scripture, you are [standing] on nothing until you uphold [the law of] the Torah, the Gospel, and what has been revealed to you from your Lord." (Quran 5:68)
"The People of the False Scripture".
OUT OF CONTEXT. The Quran does not comment on the authenticity of the scriptures while saying "ahlul-kitaab", this would mean it neither labels the books "true" or "false".
A comparable example in the Quran would be "People of Lut"(pbuh) (54:33), here the Quran does not talk about nature of Lut(pbuh) or the nature of the people of Lut(pbuh). It does not necessarily mean that the people were good or bad neither it means Lut(pbuh) was true or false. It just simply means "People of Lut" (pbuh).
The Torah is to judge the Children of Israel. (5:43).
FALSE. Let’s firstly read the entire verse...
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. (Quran 5:43)
This verse actually talks about the Torah being insufficient at the time. It says that the Jews would come to the Prophet Muhammad (pbuh) for judgement when they claimed to have the Torah that guides them.
The Christians are to judge by the Gospel (5:47)
FALSE. Let us firstly look at the verse...
"And let the People of the Gospel judge by what Allah has revealed therein." (Quran 5:47)
In the present day bible, there is actually no way to find out what Allah has revealed in the gospel to Jesus (pbuh). There are actually four narrations of the gospel in use today (namely Mark, Mathew, Luke and John) but not the original gospel. To add to these four narrations are books and letters of Paul which ARE NOT gospels even as per Christians.
This inconsistency of text has given rise to many problems to interpret basic theology. One example of this inconsistency would be, Jesus says "Keep the law", while Paul says "Nail the law to the cross". I call this a "fundamental problem" since this contradiction within the bible itself has closed a lot of good practices of the Torah (like keeping kosher, not making idols etc.) to Christians.
If Arabs doubt Muhammad's message, they are to appeal to
the People of the Scripture.(16:43)
OUT OF CONTEXT. Let us read the verse...
And We sent not before you except men to whom We revealed [Our message]. So ask the people of the message if you do not know.(Quran 16:43)
This verse tells us that Moses (pbuh), David (pbuh) and Jesus (pbuh) were men. And the Arabs could confirm THIS FACT from Jews and Christians too. It does NOT say anything about "validating" any scripture in its current state.
Muhammad himself is in doubt, he too should ask the readers of
the previous Scriptures.
Both the verses you quote talks about confirming the coming Muhammad (pbuh) which is mentioned in the Gospel and Torah. Does not mean that that entire Bible or Torah is valid or authentic.