Are there codes to insert Android emoji? Just like how in HTML you would use &amp;#36; to display a dollar character (&#36;), do Android emoji have some sort of code that I could type in a text field (and if the app supports emoji it will display it), rather than browsing through the list of all emoji?
Yes
There are Unicode character codes for standard emojis. There's a complete list available here, but some common ones include U+1F601, which is . The actual representation of the emoji might differ across platforms though, as seen below.

Additionally, simply entering the Unicode codes into the app might not necessarily display the character since it assumes you want to enter the literal code and not the emoji representation.
Several services including Telegram, WhatsApp, Slack, GitHub, etc do allow for this fortunately, through a different list of character codes, such as :smile: for . Again there's a complete list available here.