How can I set it so only one of the optional products may be purchased?

Question #2: Users will use my form to register for an event. They must pay registration fee and may also buy one-year or multi-year membership. I created a product field with one product called registration and other products called membership.  Registration is set to "required".  The others are optional.  How can I set it so only one of the optional products may be purchased?

On your form the option 'Users Can Select Multiple Products' is set to No, which means that your users will not be able to select two products - Registration and Membership.

You can set 'Users Can Select Multiple Products' option to Yes. Considering that Registration is required and selected by default your users will not be able to un-select this option. They would only need to select one of the Membership options and submit the form. However, it is not possible to force them to select the Membership option or to prevent them to select multiple membership options. 

As a workaround, you can add all of you membership options as product options of one product and enable special pricing. Then you would be able to set this product to be required as well.

What is Special Pricing

Here is a demo form: https://form.jotform.com/80143298751964 

Feel free to test it and clone it. 

Hope this meet what you are looking for.