How come it seems like all Lucy's Spirits can summon themselves? When Loki is one of Karen's spirits and he switches with Aries, he says he is not an ordinary spirit so he can switch with them and summon himself on his own but other spirits seem to have no problem summoning themselves.

If this was the case just for the Zodiac Keys, I would be "ok" but Horologium will show up without Lucy summoning him too whereas he is a regular spirit.
It is mentioned in the series that since Lucy and her spirits share a strong bond, the spirits can summon themselves without Lucy initiating it. It doesn't matter if it is a zodiac key or a silver key.

Moreover as her magic power increases, the time that they can remain in the real world also increases.