Encode text to Base64 online, emoji included

No, and this matters. Base64 is an encoding, not a cipher. Anyone can decode it instantly. Never use it to hide a password, a token or anything else that needs to stay secret.

It swaps plus for minus and slash for underscore, and drops the trailing equals padding. Those three characters have meaning inside a URL, so the standard alphabet gets mangled when passed as a query parameter.

Base64 represents three bytes using four characters, so encoded output is about a third larger. That overhead is the cost of making binary data safe to put in a text field.

Yes. Text is converted to UTF-8 bytes before encoding, so emoji, accented Latin, Cyrillic, Arabic and CJK all encode and decode cleanly.

No. Encoding happens in the page, so nothing you paste leaves the browser.