Script the indexes, then drop them and the clustered index and create the clustered columnstore

If you have a backup of the DB before you removed the compression, you can restore that somewhere and check what indexes were compressed and what compression they had.

Because a count(*) is asking how many rows are in the table, not the contents of the rows, and hence that can be answered with any index, as all non-filtered...

Maybe. Depends on the predicate, the index key columns and the rest of the query. It's not as simple as you're making out.
There's a lot of posts on indexes...

It's going to have to scan something, since there's no filter on that query. Optimiser calculated that a scan of the table will be the most efficient.