Dynamic Taxonomy Filter dropdown with childs in Drupal 7 For a while I'm trying to sort this but no luck

I need exact same filter like here http://www.pets4homes.co.uk/

I have pet types as taxonomy terms and breeds as child`s. I want to accomplish same thing like on that website, when I select dogs I want just dogs breed to be on second dropdown. Location is not needed but if you know the way to add it even better, but now I'm focusing on dropdowns.

Already I tried with Simple Hierarchical Select module and Hierarchical Select but I just can't set it up to work like I want, or I don't know how to set up it properly.

With SHS module  I have one dropdown with pet type and when I select it than other dropdown shows up, but I need it to be all the time there, and that is not possible with this module as I read it in issues of that module.
You need to use Ajax and the drupal form api to build this type of search form.

I recommend looking at the Taxonomy Edge module to make "finding the children terms of some parent" really easy, because the SQL can be annoying. Then you can just focus on the Ajax needed. Drupal has great support for Ajax!

I recommend looking here on drupal.stackexchange to find ajax examples of which there are plenty. I'm nearly positive there is an example of this type of ajax "update the form with some new options or div" in the Drupal 7 Developer Examples module.