Salaam mia,
You asked:
There was an incident in the Quran in which the Jews were told to judge by the Torah, because the Torah contained "the plain word of God." Does this mean that the Torah was intact and uncorrupted at the time of Prophet Muhammad?
I am of the opinion that the Torah (and the Gospel for that matter) were not (fully) corrupt during prophet Muhammad's (pbuh) time. And the evidence for this (as you've alluded to) is the fact that Allah has instructed each religious party to judge by their respect scriptures:
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It was We who revealed the Torah (to Moses): therein (Arabic: Fi-ha) was guidance and light. By its standard have been judged the Jews, by the prophets who bowed (as in Islam) to God's will, by the rabbis and the scholars: for to them was entrusted the protection of God's book, and they were witnesses to it: therefore do not fear mankind, but fear Me, and sell not my verses for a miserable price.
If any fails to judge by (the light of) what God has revealed, they are Unbelievers."[Quran 5:44]
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Let the people of the Gospel judge by what God has revealed in it. If any do fail to judge by (the light of) what God has revealed, they are those who rebel".[Quran 5:47]
So it would make no sense for Allah to instruct those religious groups to judge by their respected scriptures if, indeed, they were corrupted. But I do acknowledged that their are Quranic verses which allude to partial corruption of God's other scriptures. But I would venture to guess that the corruption being spoken of (with respect to the other scriptures) is
verbal corruption and NOT textual corruption. In other words, God's previous scriptures were not corrupted textually whereby mischievous individuals would, among many things, erase some verses, add their own man-made writings into it, etc. But rather, they were corrupted by way of mouth. Meaning mischievous individuals (scholars/learned-men) would decontextualization verses and give false meanings to them so as to mislead the public and bend them to doing their (the mischievous individuals') will. This (understanding/explanation of mine) would explains why, on the one hand, Allah instructs the Jews and Christians to uphold and judge by their given scriptures while, on the other hand, speaking of the partial corruption those scriptures have underwent.
You also asked:
why did the Bible and the Quran need to get revealed if the Torah was still intact
It was never God's will for us all to be one people and practice the same religion, as evident by the following verses:
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To every People have We appointed (different) rites and ceremonies which they must follow: let them not then dispute with thee on the matter, but do invite (them) to thy Lord: for you are assuredly on the right way. If they do wrangle with you, say, "God knows best what it is you are doing. God will judge between you on the Day of Judgment concerning the matters in which you differ."[2:67-69]
"....
If God had so willed, He would have made you a single people, but (His plan is) to test you in what He has given you: so strive as in a race in all virtues. The goal of you all is to God; it is He that will show you the truth of the matters in which ye dispute” [Quran 5:48]
The scriptures of God (Quran, Bible, Torah) are simply books which God uses to communicate his commandments to us. And sense God has made us different people (religiously speaking), with slightly different rituals and ceremonies, it would be appropriate for God to then sent down (for each religion) its own scripture laying out the rites and ceremonies they are to uphold. Hence why God needed to send the Quran, Bible, and Torah for every member of the Abrahamic religions.
You also asked:
when was the Torah actually changed?
This is a good question, and one which I don't have the answer to. Perhaps other forum members could answer it.
Peace.