You can add "female" or "male" before "cousin" so it becomes "my female cousin" or "my male cousin". You can say "my older male cousin" to describe a male cousin who is older than you, and "my younger male cousin" to describe a male cousin who is younger than you.

In English a cousin is the son or daughter of your Uncle and Aunt. We don't distinguish the difference as you described.

As already mentioned, most English speakers don't routinely provide gender or age hierarchy when referring to cousins. This information, when relevant, is usually provided indirectly by referring to the cousin's name and age relative to the speaker. For instance, "I have a younger cousin Katherine, who lives in Florida."

Keep in mind that for English speakers this kind of information isn't important. It will seem very strange if you try to be that specific. I couldn't even tell you which of my cousins are older or younger!