1. There are several different ways. One is to use a free forwarding service and forward the ones you want to save to the new address. This also helps to get people caught up if they are still using the old address. However, to move a large amount of emails, this is tedious.
Another way is to copy them to a hard drive or removable drive. While these instructions refer to Outloook Express, something that was 86'd by Microsoft for email, from your query, I'm guessing you use the domain provided by your ISP. This is a good example of why that is some times problematic, but the instructions will likely work regardless of your domain or client. It's pretty close to what I would have replied, so I'll be lazy and just link some instructions:
http://www.steve.maurer.net/tutorials/email_tutorials/saving_email.htm
2. If that doesn't do it, or you are looking for a free email forwarding service, Google is your friend.
3. I know it is a bummer, but if you're going to insist on trading child p0rn via email, the government will get you. Changing ISPs won't help.