Pick up ability

No, the pick up ability only gets you items out of grass as far as I know. The higher the level of the Pokémon with the ability though, the better the items it will pick up.

 

I'm pretty sure that there's no relation between the ability Pickup and the chance of encountering shinies. There are only a few valid methods to increase the shiny rate, but the internet is filled with other false methods. The only thing the Pickup ability does is, well, picking things up after battle.

 

No, it doesn't affect shiny rates at all. I don't think any Pokémon ability can do that, only the Shiny Charm (given to you when you complete the National Dex) can increase the chance of encountering shinies in the wild.

What Pickup does is pick up some items after battle and the items are affected by your Pokémon's level (the higher its level, the better items you'll get)

In Gen III and IV, pickup can get you rare TMs such as Earthquake but in Gen V and above, it only gets you regular items like potions and Heart Scales.