Google Search by Country is close to what you are looking for as it allows to:
Change Google search results to show only selected country's results.
To use it, you must:
Enter your search into Google. Once results have been loaded, use the extension's icon/button to select the country you would like the results from
You can use either of the following two userscripts to make Google search bar unsticky. Userscripts were tested on Firefox v86 with Greasemonkey addon.
Note: if you don't know about userscripts, you can get started with them here.
Following userscript keeps the search bar fixed at the absolute top. This means that if you want to type in the Google search ...
I don't know of a dedicated option to make the settings stick, but a workaround is to change the URL instead of the search field.
Suppose you're searching for images of red flowers:
The search term, 'flower', appears both in the search field and in the address bar. If you change it in the address bar to e.g. 'shoes', Google will remember the settings ...