I enabled canonical partners in software&updates and when it updated it saved everything into /boot partition which has only ~200MB and I can't see the partition now but remember making it when installing Ubuntu 14.04. So my question is if I can change the partition/folder/place that software and updates want to save stuff in?
First of all, you can try to clean some downloaded packages by running commands
sudo apt-get clean sudo apt-get autoclean
It looks like you mix up / and /boot directories. Software is not installed into /boot directory. But it could run out of place because of kernel image and headers update. You can remove old kernels by using synaptic.
Generally speaking it is not a good idea to make a separate /boot partition. This has no benefits for desktop computers, but may cause problems like you have.
There is no good way to keep kernel images anywhere ele, but /boot. I suggest to merge /boot with /, but it is not quite easy for a novice. Maybe it makes sense to reinstall the system, keeping your /home partition. I guess you have it separate. You can remove /boot partiton and increase / in gparted. You can do it after booting from LiveCD.