Loading Dropbox on start-up when my Dropbox folder is on separate partition

Yeah.

You would need to mount your partition when your system starts.

For detailed guide refer Automatically Mount Partitions in Ubuntu.

If your have external NTFS partition then ntfs-config will ease your task and let you mount all ntfs partitions easily. To install ntfs-config, open terminal and type;

sudo apt-get install ntfs-config

I edited the /etc/fstab file.

Step 1. List your partitions by UUID:

sudo blkid

Step 2. Edit /etc/fstab file.

To edit the file directly in terminal, run:

sudo nano -Bw /etc/fstab

Step 3. Insert an extra line in fstab; the line should look like:

UUID="01CE8D75DCC82F80"    /media/G-Drive    ntfs  rw,auto,user,fmask=0111,dmask=0000   0     0

You will have to replace 01CE8D75DCC82F80 with your UUID.

You will have to replace /media/G-Drive with the location where you want to mount.

You might want to change ntfs if your partition is a different type.

If you change ntfs than you might have to change options as well (rw,auto,user,fmask=0111,dmask=0000 gives anybody access to this NTFS partition).

Step 4. Restart and keep fingers crossed, if "Drobox Unlink-ed message" not shown than you have been successful.

You can edit your /etc/fstab file to add whichever partition you need. You can find the wiki here.