Yes, Mint uses Ubuntu's Ubiquity installer, as do a ton of other distributions. However, I believe the in-built partitioner in the Ubiquity installer is a bit different from GParted (which is another program you'll find by default on the Live CD/DVD). They're just slightly different front-ends for the same operation, just like, for example, apt-get, aptitude, and Synaptic are all different front-ends for APT.
I'm not sure you get what "defragmenting" your hard drive means. It does not create a block of contiguous free space as you seem to think it does. It just reorganizes files on the partition being defragmented, so that resizing
the actual partition will be safer and somewhat quicker as well. You might find giving this a quick read to be helpful: http://en.wikipedia.org/wiki/Defragmentation
Linux does not use the same Windows notation for denoting partitions (C:, D:, etc.). Here, it's /dev/sda# (e.g. /dev/sda1, /dev/sda2). The reason has to do with the fact that everything has to be mounted somewhere under / in Unix-based systems...if it doesn't have a path under /, it doesn't exist in *nix. You'd better get used to this soon otherwise you're going to run the risk of screwing up big-time during the installation process. For the Windows-to-Linux convert who struggles with this notation difference at first, I suggest identifying your partitions by their size/their filesystem at first.
Anyways, what I recommend is shrinking the "C:" partition (formatted as NFTS), and adding a new [logical] partition with the freed space (formatted as ext4) for /, or adding an extended partition with several logical partitions inside for /, /home, /swap. I would not recommend mixing up your existing Windows partitions with Linux...primary reason is that they use different filesystems. WIndows (XP and later) uses NFTS, Linux users tend to use ext2, ext3, ext4, or reiserFS (there's also JFS, XFS, brtfs, etc.). You probably don't care about why these filesystems exist and their differences (if you do, it's beyond the scope of this discussion here, so Google it
), but what does matter to you is that Linux is perfectly capable of reading and writing to Linux filesystems as well as Windows' ones (FAT32 and NFTS primarily), while Windows cannot
read or write to ext2/3/4/reiserFS/brtfs/etc by default (there are third-party utilities that can accomplish this, but again, beyond the scope of this discussion...it'll take me forever to explain every possible aspect of this to you).
Point is, unless you know what you're doing, keep WIndows and Linux on seperate partitions and use the default filesystems offered for each OS (NFTS for Windows, ext4 for Ubuntu/Mint). It's safe to keep a separate data partition formatted as NFTS if you want both Windows and Linux to be able to access it.