Getting Search Form onto a single line

That may depend partly on your theme. Can you post a link to your site?

Using Jupiter theme, I had a crack at it but not very responsive.

https://dev.denmarkconnects.com/events/#EventListTab

Sorry, I may have misunderstood what you want to do. Do you want to move the category dropdown from the right to the left?

Yes category dropdown on the left then dates.

Also, if I create two custom taxonomies using PODS can I show those in drop downs on the search form?

In that case you’d need to make a custom version of this template:
/events-manager/templates/templates/events-search.php

This tutorial explains how to create custom templates that are upgrade-safe:
https://wp-events-plugin.com/documentation/using-template-files/

Depends on how PODS works in the situations, you should be able to add the PODS custom code to the same template.