Is it possible to make a custom contact page, meaning that it would be called something else, for example "ask for a quote", and would look different than the default mydomain.com/contact/ page?

Or is it possible to embed a simple contact form onto a page?

Yes, you could create a Pages page and use that, with anything you want embedded via a HTML widget.

Or you could create a database and use that to hold the contact requests, and set up any custom fields you need and edit the form template (if required).

Thank you. However can I embed the standard contact form provided by IPS with a HTML widget or do I have to create my own HTML form from scratch? I was hoping there would be a way to reuse the standard contact form.