Until I have time to dive deeper into Mastodon and limit its log collection, assume IP addresses are collected and stored indefinitely (though it's unlikely). Whatever information you provide at registration is obviously collected and will only be used for communication directly related to the service, such as notifications.
Another thing to keep in mind are the privacy controls on this instance. At the bottom of the compose pane is a globe. Clicking this will reveal a dialogue allowing you to select the audience that will see your posts. In addition, clicking the three dots to the right of that will give you the option to post content that is "Local-only"; toots made with this option enabled will not federate to any other server and will remain local.
By default, users are hidden from the profile directory but this can be enabled from the `User preferences` page.