Should a research proposal contain a discussion of my past experience and achievements?
If your next research steps are a continuation of your current line of research, then a quick summary of work you've already done on the topic is a great way to work it in naturally, provide context for where your research lies in the field's line of questioning, and explain why you think this research is worth pursuing (look how fruitful it was in the past! Self citation).
The etiquette is probably field dependent. Fancy grants and fellowships require specific tailoring, but for a normal postdoc in my field, even just submitting your research statement as a research proposal wouldn't be too rude or strange.