Prepare the partition you want to use first. Write down what it's called. The Slack installer will ask for it during the install process. You can use the same swap partition that Mandrake uses. I would just put / and /usr on the same partition. I don't see any advantage in making things more complicated. One time I had /usr on a 2Gb partition which I thought would be plenty but it filled up and I couldn't install anymore software. Copying the whole /usr folder onto a new partition was complicated. Since then I always install onto just the one partition for everything. 10Gb should be plenty.
I've read other people have been able to boot several versions of linux with lilo but I've never been able to do it. I would strongly recommend that you use the Mandrake Control Centre to change the boot loader to Grub. Then when you install Slack choose not to install lilo, add an item for Slack to the /boot/grub/menu.lst in Mandrake. This boots Slackware 9.1 for me
Code:
title Slack
root (hd0,8)
kernel /boot/vmlinuz root=/dev/hda9 vga=788
Grub counts from 0 so all the numbers are one less than normal, also, it uses numbers not letters for the drive so for example, /dev/hdb3, the third partition on the second hard disk would be written (hd1,2) for grub.
/dev/hda7, the seventh partition on the first hard drive would be written (hd0,6)