I have a website running on a Mac/Ubuntu Linux server/Apache/ on my home network.
I want to:
1. Display video in my site from the cam running on my personal XP machine
2. Allow visitors to my website to message/text-chat with me via skype running on my personal machine, AND
3. Place voice calls to me on my personal machine from the site
all without having to install Skype or create an account on the visitor's machine. Simply go to my site, type some text and send it, or 'Click Here' to call me, so it will come to Skype running on my personal computer.

Possible? I doubt it, but it's worth a try.

Apart from 1. other are possible. Maybe point 1 is also possible.

The other are completely possible but they require a lot of server side coding.

The Skype road map includes possibilities such as messaging from web(and even calls). But when this will be done - I doubt even the Skype stuff knows exactly

1. impossible at the moment.

2. possible provided that you have a skype client running on your server. you must create a special skype account for the server, so you will only know that it's someone at your website that's trying to contact you; note that you are not allowed to collect, store, or use the usernames and passwords of other skype users. you must also create a back-end application that runs on your server and interfaces the web forms with your server-side skype client.

3. see #2 -- but in this case you will also have to take care of transcoding the audio which is an order of magnitude more complicated than just taking care of the chats messages.

please keep an eye on https://developer.skype.com/RoadMap as some of this **may** become easier to implement in the future.