Do Muslims believe that this book was divinely inspired? Is there contradictions between what the Quran says and what is contained in the Book of Revelations?
As Muslims, we do not hold The Revelation as sacred or as holy scripture. We hold several books of the Bible as holy, albeit we believe that a good portion of the original Scripture was distorted over time:
مِّنَ الَّذِينَ هَادُوا يُحَرِّفُونَ الْكَلِمَ عَن مَّوَاضِعِهِ وَيَقُولُونَ سَمِعْنَا وَعَصَيْنَا وَاسْمَعْ غَيْرَ مُسْمَعٍ وَرَاعِنَا لَيًّا بِأَلْسِنَتِهِمْ وَطَعْنًا فِي الدِّينِ ۚ وَلَوْ أَنَّهُمْ قَالُوا سَمِعْنَا وَأَطَعْنَا وَاسْمَعْ وَانظُرْنَا لَكَانَ خَيْرًا لَّهُمْ وَأَقْوَمَ وَلَٰكِن لَّعَنَهُمُ اللَّهُ بِكُفْرِهِمْ فَلَا يُؤْمِنُونَ إِلَّا قَلِيلًا
Among the Jews are those who distort words from their [proper] usages and say, "We hear and disobey" and "Hear but be not heard" and "Ra'ina," twisting their tongues and defaming the religion. And if they had said [instead], "We hear and obey" and "Wait for us [to understand]," it would have been better for them and more suitable. But Allah has cursed them for their disbelief, so they believe not, except for a few.
See also Qur'an 3:78.
However, when it comes to The Revelation (among other books in the New Testament) where it is clearly documented that the book was authored by one of the disciples (John, in this case), then the book is not considered to be a revelation from God (albeit that it may have quotes from revelations from God that John may have heard from Jesus) as far as Muslims are concerned.
We believe that Jesus is a prophet and a messenger of Allah and that he received divine revelations. We also believe that his disciples were pious people but did not receive direct revelations from God as they were neither prophets nor messengers of Allah:
عَنْ أَبِي هُرَيْرَةَ قَالَ قَالَ رَسُولُ اللَّهِ صلى الله عليه وسلم: أَنَا أَوْلَى النَّاسِ بِعِيسَى الأَنْبِيَاءُ أَبْنَاءُ عَلاَّتٍ وَلَيْسَ بَيْنِي وَبَيْنَ عِيسَى نَبِيٌّ
Abu Huraira reported Allah's Messenger (ﷺ) as saying: "I am most akin to Jesus Christ among the whole of mankind, and all the Prophets are of different mothers but belong to one religion and no Prophet was raised between me and Jesus."
As for the content of the Book of Revelation, there is a good deal that is common with some of the prophecies we believe in as Muslims, but again there is a good deal that we do not believe in as it directly contradicts our holy scripture (Qur'an and hadith), and a greater deal than the two previous ones that we neither confirm nor deny. If you have questions about specific verses, you may want to post them as separate questions.