I don't know your configuration so it's hard to troubleshoot the problem. What sorts of errors are you receiving when you can't log in to your imap server and what errors are you receiving when you are not able to send mail. Your server logs should have details on what the problem is.
For sendmail and other MTA's, there may be a few issues to consider.
1. Your ISP may be blocking port 25 for all but their own mail servers. It doesn't sound like that's your problem since it seems to be working some of the time.
2. Due to spam, other MTA's will check to see if you have a valid domain and will automatically reject your mail if your IP is not matched to a host name.
3. Again, due to spam, other MTA's automatically block e-mail from general user space such as the Comcast range. Per Comcast (and this is an example; your ISP may have different policies), you're not allowed to run servers on your home cable connection. Since Comcast doesn't block server ports at their firewall, you can get away with it but your mail may be generally blocked.
Why can't you get a domain though? There are lots available at pretty cheap prices. You can hit
http://www.dyndns.com to register and manage a DHCP IP address. Their DNS servers accept input from your server as to its true IP and will reconfigure the IP for your domain to point to your server again. It's pretty quick. I used it until I started using a colocated server.
As to your request for howto's, a very fast google search turned up:
Linux Home Server HOWTO
You might want to consider searching for the information next time. Try searching on "howto +sendmail". In this case, that was one of the first items that popped up. It doesn't have imap info but I'm sure another seach would turn up an informative howto.
Carl