Are pictures of the Elemental arrows (fire, ice) required for the full compendium?
Solution 1:
Yes, they are required; the five special arrows are entries 349 through 353 in the Compendium (354 to 358 in Master Mode). You don't need to photograph them while they're flying through the air, though. Among other options, you can photograph them while they're lying on the ground or a table in a shop that sells them. The "Slippery Falcon" shop in Tarrey Town would be ideal for this.