Does Remote Keyboard have Emoji support?

Emoji's are indirectly supported via the "replacement" feature. You'll have to provide the app with a mapping of ASCII smilies (or other text shortcuts) to the actual icons. After that you can use the INSERT key to swap out the ASCII string with the desired Emoji. Some general mappings are linked on the download page.

Please keep in mind that Emojis are still text symbols (even though they may look like icons). They will only display properly if the displaying device has an according font installed.