I highly recommend you read this and this.
- Would you host on YouTube and simply insert the embed code in to the page. Would any alternative site (or even self-hosted) be preferable?
Self-hosted is preferred but keep server costs in mind. If not, youtube is usually the way to go.
- Should I be padding out either side of the embed code with a description of the video and an annotation on OUR site?
Ideally yes. A part of video SEO is it's context.
- Does Google actually look at the link and ascertain the relevance of the video file to the actual page? If it does this does it gain anything useful that could help on-page SEO?
Videos can reduce bounce-backs, increase backlinks (people tend to link to them more), show a rich snippet in SERPs, improve conversions (if done right). Google can't watch and categorize the video but does it instead on the page's content, surrounding text, and backlinks.