Which countries have a specific method for folding their national flag and what are those methods?
Official guidance on the correct form for folding the Irish flag is set out in section 9 of this publication of the Irish government [note: .pdf document]. But it’s not mandatory.
I found that there is a formal method to the ceremonial folding of the Canadian flag:
Australia’s flag instructions include useful diagrams - https://www.pmc.gov.au/resource-centre/government/australian-flags-booklet-part-two on pp. 22-23.
I assume you already know, but for the sake of completeness, there’s an official way to fold the US flag. It ends up in a right triangle, with only the blue-and-stars part showing.
There’s also an official way to fold the Ohio flag, which is relevant because it’s an unusual shape.
Which leads one to wonder how you fold a Nepalese flag.
The technique for the blue triangle with stars on it is pretty easy:
Rewatching Parks & Recreation I notice the Indiana state flag on the wall is folded so that it is a right triangle with the torch showing. Do any US states (Ohio already mentioned) have any official procedures for folding their flag?
California should have done better. Apparently, you fold it into a triangle that leaves just the “CALI” showing, thereby advertising that California disobeys one of the rules of good flag design (“No writing on a a flag”). Should have figured out a way to fold it with the bear showing. Of course, a lot of state flags are just a misused state seal on a solid background. Not much clever you can do with that.