What I'm actually trying to do is install a new distro in place of Linux Mint on a machine dual-booting Windows without wiping Windows from my computer (partly because I've got backup data there from this Mint install), and I have no idea what I'm doing.

All I see on the other distro's installer is a partition table, with a couple NTFS partitions (Windows 7) and a couple other partitions (Linux Mint, one would imagine). What can I do to install this distro over Mint without losing Windows 7?

I just did it myself buy downloading a free partitioning software and ran the software in windows.
I used EASEUS free Partition Master. You can use this to unallocated you old Linux OS and when you reboot with, lets say Ubuntu live CD, You will be given the option to install along side Windows

Linux makes at least two, usually three partitions when installing so the others you seeing are probably your Mint.
If you decide to go this route be SURE to defragment your windows first and always backup.

You can also use your linux live CD When you install to manipulate you partitions. Detailed info here;
https://help.ubuntu.com/community/WindowsDualBoot

If you Google "dual boot windows and Linux" you will find many tutorials that give you step by step instructions.
;
Hope this helps a bit.

Simple as that: Boot into Windows, remove all non-Windows partitions, start from your Linux install-CD and let the installer use the free space.
Don't forget to backup your data first.

Be sure you don't remove any partitions that may contain OEM windows data or the small windows 7 partition (usually 100m).

I agree that a good backup is the cure for opps.

Personally, I'd see if a virtual machine would work. It makes it easy to change, store, copy and add and remove linux and can be run W7 and linux at same time.

I usually:1. Press Window key, 2. Right Click on Computer, 3. Select Manage from the drop down menu, When the Window opens Click on Disk Management, 4. When the partitions show delete the volumes that are not Windows, 5. Insert the CD or DVD of the Distro you wish to install, and 6. Reboot.
Cheers. Ray.

Sorry it took me so long to reply. @_@ Anyway... I did it via the distro's installer's partition manager. All I did was format the ext4 partition from mint, set it to root or whatever, and leave the swap alone, and it handled itself from there just fine. Thanks for all the info! ^_^