> Does a client push a note to relays after the initial publication? I don't think it does.

I know of no client that does this, but it can be made when/if this becomes a problem.

> So that means at some point, the older notes and other stuff will start to disappear from the internet unless permanent storage is incentivised in some ways.

Yes, true of all human knowledge and output.

>The question is, how do we do that? Because most of the relays are offering their services for free. For how long they would provide this service for free with the incremental cost of storage?

I know of no solution that will work on an infinite timescale. However, notes themselves, if only text, are incredibly cheap to store. The phone in your pocket can easily store millions of them. Media is a bigger issue.