Yeah it is not very necessary to get a new passport since you still got one that is valid. You may still have 6 or 7 months before you apply for your ILR, but if I am you, I won't take a risk of applying for a new passport, which will takes maybe 2 or 3 months for the result to come...and BRP changing name as well may also take quite a while and it's unpredictable how long it's gonna take.
There's a lot to remember indeed with the Life in the UK book. My only advice for you is, keep reading and understanding the entire book. Bear in mind that every detail in the book is important. Don't miss a thing. Make sure to understand each important events. Even if you finish the book, keep re-reading and reviewing it. Don't rely on practise questions and questions! I strongly discourage the practice answers and question book or online test. None of the questions there will come out in the exam. Actual questions are pretty tricky and misleading. What I did was, after booking my test, I took a week off from work just re-reading and re-reading and understanding the whole life in the UK book for the whole week...gladly I passed the exam and felt that a week off from work just to study was really worth it!
One day service (Public Enquiry Office)officer will also check your documents but in case you are have a missing document, the office won't provide it for you....
The Settlement Checking Service will check everything first, making sure that your documents are complete and not missing anything before the day of your application, it may be by postal or by the public enquiry office (one day service).