How can I specify limit on input number? I have a Zip Code field, and I want to (a) require 5 numeric digits and (b) greater than 01000

This could be achieved with Short Text Entry (Text Box) field. Inside of field properties need to set Validation to Numeric and you also need to Set Input Mask option to ON and add ##### inside of Input Mask field.

This settings will force your users to enter 5 digit number. 

To prevent them to enter number less then 01000, you will have to use conditional logic. If number is equal to 01000 or less, you can show your custom massage that will inform your users that the zip code must be greater then 01000. You should also hide submit button to prevent them to submit the form is Zip code is not correct. 

How to Show or Hide Fields Base on User's Answer

You can set your condition like this.

Here is my demo form: https://form.jotformpro.com/71174409372961 

Feel free to test it and clone it. 

Hope this will help. Let us know if you need further assistance.