Good feedback, thanks Nico. The comments module is kind of weak right now, so it needs some attention very soon. The long-term plan for the 'Comments fieldtype improvements' is to convert the comments system over to use pages instead. The goal with converting them to pages will be to make them really accessible and searchable from the API standpoint. It'll also mean that you'll have a 'comment' template that you can add whatever fields you want to.
However, we'll likely make improvements to the existing Comments fieldtype before this. In fact, I've already built a new ProcessComments module that adds a 'Comments' option to your admin top nav, and lets you manage comments separately from pages... just haven't committed it to the core source yet. I will take your suggestion to add a website field and keep track of the email/website in the user's session.
Execute comment saves before rendering a page because it's problematic if you want to show e.g. "5 comments"
Some of this is at the discretion of how you utilize it. I recommend doing your comments output generation and processing before you send any output. When a comment is added, make your page redirect back to itself to both prevent reload-duplicates and start with fresh DB-loaded data. This also ensures that the user will see their comment rather than a "thanks" message (great if you aren't using a moderation queue).
Add a "$comment->avatar" working with gravatar (or is it already?)
This is not already in place. I thought I saw this on your stadtpirat site? Do you know how to do it? I'll add it if you tell me how....