Is there any way to make a backup from download mode? I have an Idol 3 phone with a broken power button and a broken touch screen. I am only able to boot into download mode by holding down the volume up and down buttons while plugging it into the charger.

Is there any tool that will allow me to make an image of the internal memory on my phone instead of flashing it? Can QFIL do it, or is that just for writing data?
Unfortunately, "Download Mode" is just one direction (as the name suggests: Download data to the device to be run there) – so no, you cannot use it to read data from the device.

For comparison, take a look into our fastboot tag-wiki – as fastboot is the equivalent to Download Mode. There you'll find a link to can fastboot be used to backup device? Though fastboot isn't identical to Download Mode (especially with Samsung devices, where the two are not compatible and one has to use tools like odin instead), both serve the same purpose.

Speaking of QFIL: Especially not. QFIL is a tool to communicate with the device while in emergency mode ("Qualcomm 9008"), and only intended to make the device operable again (AFAIK) – e.g. to flash a bootloader. And of course it's for Qualcomm devices only (as others are unlikely to enter a Qualcomm emergency mode).