I used to have a dual-boot system consisting of Ubuntu 14.04 and Windows 8.1. Now I have deleted the partition for Ubuntu but the GRUB loader is still there, only that it has more limited commands now. Can I install Linux Mint using the GRUB from the uninstalled Ubuntu or do I have to remove GRUB completely and then install Linux Mint?
GRUB is just a bootloader, it should not have any impact on what you can install and what not.
Chances are that there are still some valid boot-configurations lying around, which is why they show up in the menu though you deleted the Ubuntu-partition (you probably did not delete the bootsector, thus they are still there).
When you install Linux Mint it should override your installation of GRUB, thus you will have a fresh GRUB.
If you want to get rid of the remaining ubuntu-bootsectors, check out this.
No , you don't need to do anything extra, just create a bootable pendrive of linux mint and you can install. but this time option
Alongside windows boot manager option won't show up. You will have to select Do
something else option and then identify your old ext4 partitions ,format them and give a single partition for / directory i.e root ( no swap needed for newer system , you can do few extra partition for boot and home ,but not necessary), and you are done ! Old grub is overridden by newer grub :) grub menu back .