Do you prefer typing emoji instead of stopping to choose them from a picker?
You can do this when writing a post on Mastodon's web interface by typing a : (colon) and then a keyword without a space. This will automatically bring up emoji with that keyword in their description.
For example if you type :sun it will show you emoji with the keyword "sun".
(Note that if there are many matches for a keyword, you may need to go to the picker to see them all.)
@feditips somehow I'm limited by what's available on this server. Can I access other server emoji's in the same way?
No, but you can ask your server admin to add emoji from another server.
The admin's custom emoji settings include an option to clone emoji from other servers.