If I want to send out HTML emails via MailChimp, would they host images for me or do I have to do it on my own server?
From MailChimp Full Feature List :
Free Image and File Hosting
Upload images and files into your campaign without having to worry about extra charges.
You can upload images in designer view. Than keep URL and use it in advanced view with your HTML. It's not as easy and sleek, but it works.
Check details here.
You can also use Flickr to load images. And edit them with Picnic. Cool features.
The "big famous competitor" that is an alternative to mail chimp is Constant Contact. They also offer the "click here to view this as a web page" option, I recall.