Better Way To Install Ubuntu 15.10 Without Making Partitions
Solution 1:
Is there way to solve this mounting.
See answers of Ubuntu 14.04 not booting after error message. /tmp could not be mounted
Is it even possible to install ubuntu 15.10 in the same way as wubi did, but manually, or any other software.
I installed 15.10 with Wubi without any problems. There is no official Wubi solution (besides to remove it from isos) but community supported versions still exist. See solutions of other Wubi users like this one.
Solution 2:
If you're not doing anything terribly resource intensive and just want to play around with it (which is implied by the fact you're wanting to use Wubi instead of giving it its own partition outside of Windows), you could just install Ubuntu in a virtual machine.
I use Oracle VirtualBox (www.virtualbox.org) to create and manage virtual machines, but there are others out there that are free.
If your concern is for your Windows partition, you can always just use GParted Live to resize your Windows partition back to full if you install Ubuntu and change your mind about it later.
You can find a little more information about the state of Wubi at this askubuntu page: Windows installer for Ubuntu 14.04 LTS onwards