yes, updates performed by triggers will fire other (including "recursive") triggers just as any other update would.
since you're dealing with a tree structure (no cycles) this means your job is pretty simple. doing something like
update table set weight = weight + (new.weight - old.weight) where id = new.parent_id
should cover you (allowing reparenting of nodes would complicate things some, but you could check for and handle that if you needed to). top-level nodes should have parent_id = null so an update there will result in zero rows updated (by the trigger) and no subsequent triggers firing. similarly for your ingredients/mix tables, your "top level" incredients will not exist as components in the mix table.
