As the question reads, do these games still require Flash player?

Also is it possible to run them with WINE?

I uninstalled Flash player last year and all Samorost games run smoothly.
You can download Samorost 1 for free on Amanita website if you want to check.

No idea about WINE though

Not sure, but iirc. it is kind of a stand-alone flash distribution, so it uses Flash technology (*shudders*) but it probably does not require any external Flash player installed.
We all know that Flash was announced dead long ago and it already is dead.

You might check the Wine databases (or proton) if there is an entry. It might work, since it is "static" / standalone and thus should "just" require the W32 API. However, I do not know if the included Flash does strange things in terms of trying video acceleration that might end up giving artifacts.

The game is free-as-in-beer now, so we could simply try and hope.

Just downloaded it and tried with whatever WINE I had currently here (Gentoo, =app-emulation/wine-vanilla-6.11 outdated but I didn't yet compile a newer one) and it seemed to work fairly well. Just messed with the game for 3 minutes but so far it seemed okay.
Hope that helps.