How can I delete a mount or minion (if possible)?

[Zetsumei_Tsunarashi]

Once you have it listed in your mount or minion list, it's there forever. No way to remove it.

Just because you have it though doesn't mean you need to use it. I do similar things to what you're saying. For example, I only summon my griffin when playing as a gatherer or crafter, while as a battle class I only use midgardsormr.