How to create a redirect form?
To get the form with a suggestions list like that you need to create an exposed filter in your content Views and use Exposed form in block option. Place the block in a region of your theme. 

To get the autocomplete suggestions list for taxonomy terms use the Has taxonomy terms filter and set it to Autocomplete. 

The Global filter doesn't seem to work with taxonomy fields, I'm not sure if there are combinations you can try to make it work and have all the vocabularies terms searchable in one field. 

The default Taxonomy term Views contextual filter needs to be removed since it won't work with only exposed filter output unless you use Views Filter Harmonizer. 

Query Parameters To URL can help with the SEO side of things. 

To get the Term name into your content Views instead of the TID you have to add a Relationship to the related Taxonomy term field. Then you'll be able to add Fields, Filters and Contextual filters based on the "Name", but the widgets might not offer autocomplete or other options you have under "Has taxonomy term".

Alternatively you can create a "Taxonomy term" TYPE of views where you can get the Name filter for all vocabularies, but no autocomplete. 

(Add Relationship to content so you can show nodes in the results.) 

Search Autocomplete might help partially, maybe you can get it to direct to a specific views page with the result as argument. 

Using a more powerful Search API or other search features related to Views might allow you to achieve everything you're asking about.