I need to know if this software can do the following

embed into a website

save and resume from

View and download

edit form fields after submission 

indicate to the user that the form has been submitted in their online portal

Yes, forms can be embedded: https://www.jotform.com/help/67-Which-Form-Embed-Code-Should-I-Use.

---

Yes, form users can save and continue later by Autofill: https://www.jotform.com/help/227-How-to-Enable-Auto-Fill-on-Forms.

---

Can you add more details of the View and Download?

---

Yes, edit submission is possible: https://www.jotform.com/help/40-How-to-Let-Users-Update-Their-Form-Submissions-at-a-Later-Date.

---

If you wish to be notified when someone submits your form, you have to create a notification email: https://www.jotform.com/help/25-setting-up-email-notifications. If you wish to notify the form submitter, you have to create an autoresponder email: https://www.jotform.com/help/26-Setting-up-an-Autoresponder-Email.

I hope that helps.