Your output of fdisk -l shows that /dev/sda6 is a FAT16 filesystem not an EXT4 as you said. Try formatting the partition before installing Fedora on it.

I had the same problem some time ago. Fedora tries to group the /, /home, and swap partitions on an extended partition and creates a separate primary partition for /boot. It might not install because you already have 3 primary partitions. It did not install for me. I had to switch the disk to GPT. Or you could erase the extended partition, install Fedora using all available space, and create as many logical partitions as you need inside the Fedora extended partition. Then copy your files in the original /dev/sda3 and /dev/sda5 to the new partitions.

I assumed your /dev/sda1 and /dev/sda2 have Windows installed on them.

EDIT:

Easiest way is just like "hedayat" suggested: Remove /dev/sda6 and let Fedora do the automatic partitioning on the newly unallocated space.

I'd suggest let Fedora automatically partition space for you (You can select 'standard partition' rather than 'LVM'). However, remove /dev/sda6 before that (or you can select it to be deleted in Fedora installer 'Reclaim Space' dialoge).

You can also do a custom manual partitioning: You might create a separate /boot, but not required. But, remove /dev/sda6 and create at least two partitions in it: one swap partition (formatted as swap) with sufficient size, and one / partition of type ext4 (which should become /dev/sda6 and /dev/sda7). If you are unsure about partition sizes, you'd better go with automatic partitioning as I said. Just delete /dev/sda6 and let Fedora do the partitioning for you.

To install fedora 20, please keep at least 30GB disk space partition on ext4 filesystem format.