Hugo generates static HTML from a set of shared templates, so the script only needs to go into one partial — the one that renders <head> — for it to appear on every page the build produces.
It can, if the theme is a git submodule replaced wholesale on update. Overriding the partial in your own project's layouts/partials/ directory, rather than editing the theme's copy directly, survives theme updates since Hugo checks your project first.
Yes — since the head partial is typically shared across all language versions of a page, the script tag is included regardless of which language a visitor views.
No — config.toml controls site configuration and build behavior, not raw HTML output. The script tag has to go into a template file that actually renders <head>.
No — adding one static script tag to a template has no measurable effect on Hugo's build time, which is dominated by content processing, not head-tag size.