Security, updates, servers, slackware and working remotely

Hi there!
I've been using Slackware for a couple of months now... it is the
version 10.0 if I remember right. At this point in time I'm sure it is
out of date, insecure, etc...
I plan on securing this computer (after I reinstalled the system) and
I was wondering what is the program that takes care of the firewall?
Also, I will be working remotely on my computer. I'm thinking on doing
all this through an apache2 webserver using PHP scripts and a secure
login script I developed long ago. What else should I consider for
security? Updating apache of course, but should I go as far as updating
the kernel or anything else? (If you read more, I'm also thinking on
using an SSH server, so I would update this as well)

These were my most important questions... I also have a strange
question, one that is more twisted... I'll be working from work, where
I don't have any permissions to install any kind of software (windows
2000) such as PuTTY. I was wondering if there existed a secure SSH
client that works on Internet Explorer? If possible not using java.
I'm thinking on something more like Flash (I think java is not installed
on the PCs at work).

About my work, very simply, my supervisors don't care what I do when
we got some idle time. But it's impossible to even think about
installing or modifying whatever is already in the computers. I can
change my background picture at least!

Thanks for all the help,
Turner