I am working with the weather plugin. I want to expose the properties popup to the user.

Likely it is straightforward to modify weather's default.htm to include a gear icon which will show the properties from defineProperties.

Will the properties still be available via $this->property?

How do I refresh?

Yes, values from defineProperties are available via the property() method. Refreshing the page should be done via the AJAX framework.

I am having trouble making an ajax call. copied from the docs into a page

<button data-request="onLoadContent">Go</button>

works, but

<a data-control="popup" data-handler="onLoadContent" href="nojavascript...;" class="btn btn-primary btn-lg"> Launch Ajax Form </a>

doesn't :/