Sorry about the late answer, there are many more tickets than you can see in the forums, that I had to answer first.
1. It means that the theme is not using the searchform.php method to display the search bar, thus it's only possible to replace it manually as stated in the documentation. To do that, you will need the search PHP shortcode. I suggest consulting with the theme developer in case you don't know exactly where to put the shortcode.
If you want, I can take a look and investigate where it possibly is located.
2. The “&post_type=product” does redirect the results to the WooCommerce results page, which does not support displaying any other content than products unfortunately. Ajax Search Pro passes the tags and every result to the WooCommerce shop results handler, but it simply removes them.
3. I cannot see this error message now. Did it disappear, or is it visible on a specific page?
4. The search speed is ultimately determined by the server speed, but there are many performance tweaks you can try. Please check this chapter in the documentation: https://goo.gl/Rn6lG6Best,
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