Create the form you want to host on your wordpress.
Install the Mautic plugin for Wordpress and configure it to point to your mautic instance.
Embed your form into the wordpress page you want it to show up
Create the email you want as an auto responder, make sure you create this as a campaign/template mailers
Create a campaign, open the builder and start the campaign with a form being submitted, then action send email and then choose your email.
Mautic uses segments instead of folders, and segments are very clever, they can be static or dynamic. There are so many ways to do what you want, maybe what i would do is give each new contact a tag (you can do this inside your original campaign above) and then create a segment with a filter based on tag includes.
Weekly mailer based on their folder, very similar, create segment, either create a campaign that will send out the emails or segment mailer and send to that segmetn.
PM me if you want to jump on a 30 minute call and i can show you how to do everything.