Will the forms developed with Jotform be mine to keep?

Yes, you can keep the forms in your account. We do not delete forms unless the forms you create violates our Terms of Use (https://www.jotform.com/help/8-Terms-of-Use). 

You can also export your whole account data including the forms if you would like to and keep a local copy of it.

https://www.jotform.com/help/374-How-to-Export-All-of-Your-Data-at-Once