Compound Eyes Accuracy of its attacks increases. <-- how much?

By 50%. Meaning, a Butterfree's Sleep Powder has an accuracy of 100%.

Rain Saucer When Raining, it is healed little by little. <-- And yet again, how much?

Probably heals like Leftovers does; 6.25% each turn.

Find Stuff Picks up items. <-- does it do anything in battle?

If the bearer has no item, it picks up random ones during battle.