If you were at one point already running Ubuntu 15.04 w/out any problems, then yes. You Should be able to run 15.10.
If the computer you are planning on installing supports 64 bit architecture then USE IT.
I prefer to use a USB Flash Drive when installing Ubuntu. If you think you want to try the USB method then check this out. There might be more steps involved and it can be harder for a someone not familiar with this method.
If you want to use a DVD (Some people find this simple and easy to do) then all you need to do is download the Ubuntu 15.10 iso file and burn it to the DVD.
As far as the old version of Ubuntu still being there. It's honestly up to you. When you start to install the new version of Ubuntu, it will ask you how you want to install it. You can have the option to remove everything and only install Ubutnu 15.10 OR you can install it alongside the pre-existing version of Ubuntu.