How to import an external database to D8 and create a search form?
The best and most flexible option at your disposal will be to use the Drupal Migrate module. It's included by default in D8. 

The steps you'll have to follow is as below:


Migrate your data from MySQl to a Drupal content type or an entity. You could also consider exporting your data to CSV and import from the CSV. 
If you are not expecting very large traffic, use views and exposed filter to implement your search form with filters. If you are expecting the more traffic and scale user search_api and use something like Solr in the backend. 


Some of the links where you can get started:


Migrate to Drupal 8 from MySql.
Migrate to Drupal 8 from CSV file.
Official Migrate API documentation.


I am assuming you already know what content type, entity, views, exposed filters etc are. In case you don't know any of it, plenty of resources are available online.