If the road tax hasn't been paid for ( over ? ) 3 years then the green book is invalid. The bike needs to be re-registered .. If you say there tax has been paid every year, then no problem the book is still valid.
The DLT won't accept a copy as you need to present the original book and the new owner ( you ) will need to be printed on the page following the old owner.
Yes, it is a problem to ride a bike without changing the name. You are required to keep the book ( a photocopy is sufficient ) with you when you ride the bike. If you are stopped at a checkpoint and IF the green book is checked against your license, then there could be a very slim chance that you have a problem. I got stopped once on an old Honda Dream, and although they let me go, they told me to change the name on the book as it shows the bike could be stolen.
Just work on getting the bike in your name .. Collect a certificate of residence from immigration, and if you have the green book, then its easy to change the name.