I have Amazon Prime and as a result I can leave a "free" Prime subscription with a channel.
My question now, will this subscription persist and will it cost me something at some point?
No, it doesn't cost you anything extra. As soon as you no longer have Amazon Prime, you can renew the subscription manually for a fee.
Prime Student is an exception. You only have a Twitch Prime subscription there for one month. The normal prime remains as normal.
There was something from 30 days… Is the subscription then gone?
Not really. After these 30 days, you can either renew the subscription with the Twitch user or subscribe to another.