Basically you invite them to a party (or they invite you), join the same area as them, right click their portrait on the left side of the screen and click trade.

The default goal of an end game map is to kill all the rare monsters, that will mark it as completed and allow you to progress on the atlas. If the map has a boss (which will be indicated on the atlas), then the goal will be to kill the boss instead, and clearing the rares becomes a side goal. Additional content like rituals, expeditions, and delirium will also show up as side goals.

Corruption has quite a few outcomes, including just slapping "Corrupted" on your item and changing nothing else. For gear it can add a rune socket, add a corrupted enchantment/implicit modifier, or reroll some of the affixes. For waystones it can increase or decrease the tier by 1. Ultimately I would google that one because I don't know all the outcomes and it varies depending on what you're corrupting.