Is it possible to fully customize zendesk new web widget (CSS and HTML both)? I have tried searching the web, but I am not able to find any answers.
According to this support post, it appears that the answer to your question is "no": https://support.zendesk.com/hc/communities/public/posts/204143646-Web-Widget-Customization-Options?locale=en-us
You could probably override the css and/or html using jQuery, but ZD warns against this in the post above, saying that this is not supported.
You may want to post comments in that support post to show your interest in ZD developing this feature.