I have a custom form and multiple fields. Is it possible to add a custom fields when creating and submitting new case using REST api?
The Force.com Tooling API Developer's Guide says you can add custom fields to custom objects.
If your calling code will Apex, the simplest solution would be to use Andrew Fawcett's Apex Wrapper for the Salesforce Metadata API. The documentation page has an example for adding a custom field.
You can use Workbench for testing your REST calls.