Alerts stay until they expire. Viewed alerts automatically expire & drop off after 4 days. Unviewed alerts automatically expire & drop off after 30 days.
Conversations aren't really deleted, but you can 'leave' them and they won't show up in your inbox anymore.
If you want to 'delete' a conversation, click on the conversation and click the Leave button at the top of the conversation
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Alternatively, on the conversations page, you can check the box on the conversation(s) you want to leave, then scroll down to the bottom of the page:
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Click on Choose action, Leave conversations, then click Go.
I just added this to the
What is a Conversation? FAQ post.