Does a Search Engine send a request to the server every time a new character is inserted for new suggestions in order to employ an autocomplete feature? I'm referring to search engines that look through an index and not those web-surfing crawlers.
No, there is a two configuration knobs:
min-number-of-characters that the minimum number of chars in the query before sending a request to the index.
The other configuration option is something like timeout when you type it will wait that amount of time before sending the request, with possibly a "backoff" strategy that reduce the the timeout to zero to send immediatly the request after a few keystrokes.
By the way "instant search" is different from "query suggestion"