Ok, I've got a bit more experience with download mode now. The way you're trying to do it can be made to work but it's different than what I originally wanted you to do. Let me try to explain.
A partition can contain a filesystem, or it can contain... something else. When you format a partition on Windows, you're creating a filesystem. I didn't want you to do this.
So, normally a disk contains partitions, a partition contains a filesystem, and a filesystem contains files. What I wanted you to do was create a partition, without a filesystem, with one file written over the partition, starting at the very first sector of the partition.
What you tried to do was create a partition with a filesystem, with a file stored somewhere (we don't know where) inside the filesystem.
But, I found out that if you have a fat32 filesystem in the first partition, download mode seems to mount it automatically. So it turns out the way you were trying to do it can be acceptable. You would have to change the command like:
cat /storage/external_SD/TWRP-filename.img > /dev/block/mmcblk0p17
But... we shouldn't be doing any of this stuff. If you see a storage device from the phone when you turn it on, I think something else is happening. I don't think it's running download mode (do you see the "Firmware Update screen and circular arrows and "ROOTED" at the bottom?) If you don't see this stuff, it's not download mode, and trying to cat the recovery image like such won't work. In addition, if the phone's modem partition is showing up and it's empty, more is broken than the recovery.
I think we need to restore a whole 7.5GB image of the phone. But I think that can be done if it's showing up as a mass storage device. It's easiest to do that in linux, I think, but certain disk imaging tools should be able to do it in Windows as well.
http://hddguru.com/software/HDD-Raw-Copy-Tool/
I think what I'd like to do first is have you run this, and create an image file of your phone, then let me know the size of it in bytes so I can make sure I have a matching image that is the same size. Note: I can't run that program here so I'm not 100% sure how it works, I can't tell from the screen shot if it actually does what we need or not. Maybe we need to find a different one.
I'm about to do a test here of whether cloning an image from one phone onto a different phone will actually work and that they didn't have some kind of protection to prevent that from being done (and it doesn't do weird stuff like clone the ESN over, but I doubt this is stored in EMMC). I'll let you know how it goes soon.