If you are going to do a search you either need a keyboard or voice entry. If you have a real keyboard attached via bluetooth or USB, you did not mention that, and I don't have a hardware keyboard to test with.

Works the same for me in all three browsers you mention. Depending on your choice of virtual keyboard you can exit the keyboard by pressing the keyboard symbol or by pressing in the body of webpage. I have my keyboard set up to provide me the speech option choice, but it takes up the same amount of space when clicked on from the virtual keyboard as the virtual keyboard itself.

When you invoke the search via "Next" or "Go" the virtual keyboard should disappear.

You can change your input keyboard choices via long press on setting symbol on the keyboard , or though it never (thankfully) worked for me swiping across on the spacebar. Or change keyboard by long press in location bar then select "Input method" or change through settings > keyboard.

I do have a problem in this area with the virtual keyboard not completely going away if I am looking at my Verizon email in landscape mode.

The virtual keyboard appears when the site sets focus to a text box. An example of this is https://www.google.com you will notice that you can start typing right away because the text box has focus. The same works on mobile.