My host company doesn't have curl extension enabled automatically, however, I'm able to load it using dl().

What will be the correct devote WordPress to load the extension to ensure that it might use curl for wp_remote_* functions?

I'd want it to survive the potential upgrades of WordPress code.

The first hook I understand of is init. My recommendation is always to build this like a plug-in (to ensure that it'll survive upgrades) and perform the following:

add_action('init', 'load_curl_functions');

function load_curl_functions() {
    //Use dl() to load curl

---- EDIT ----

It appears like you will find some hooks that fire before init. I suggest attempting to hook to load_textdomain rather. This is actually the hook that loads language and translation functions (the only real hook that fires earlier is muplugins_loaded that might not operate in non-mu installations).

So: add_action('load_textdomain', 'load_curl_functions'); should load your curl extension before doing other things ...