When searching for the desired SE site, the down arrow key (down) should work in order to choose the site one is looking for and hit enter to visit it, instead of using the mouse point to choose it. Would be pretty convenient.
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I've made a userscript for these purposes.
Second, explanations, all apply only if the dropdown is expanded:
First click on up/down arrows brings you to the list of all communities, further clicks navigate you up/down this menu.
Typing something immediately sets focus to the search box.
Clicking enter brings you to the selected community if any selected, otherwise scrolls you to the search box (latter is the default SE feature, unmodified)
Third, the script itself! Download it from Greasy Fork.