Search improvements and upstream fixes to SPHT

[agriyakhetarpal]

Thanks @agriyakhetarpal, this looks really good and in terms of functionality it is working well 🚀

However there can be some UI improvements (I couldn't stop myself comparing it to https://docs.pybamm.org/en/stable/ search bar):

• We should have some hover over styling property on the search button

• While search is enabled, I can still scroll the page unlike on the docs

• Some text to indicate it's keybinding on wider screens

• It is so fast that I don't feel the necessity of a loading circle but can it take time in some scenario?

• I'd personally not prefer pitch colors used in the background to keep it familiar with the wesite colors:
  

• clear button on searchbar closes it and it should also have some hover effect

• And this one is also personal - I'd rather keep the icon a little more minimal but it depends upon how you and others also think about it.

Can't wait to see it 😄

EDIT : I see most of it is from the search implemented on scientific-python-hugo-theme so I'd not mind if these points are not addressed too. That being said, I'd still encourage you to see if we can override these properties.

Originally posted by @arjxn-py in #210 (review)