How to unlock the Hunting Log in FFXIV ARR

Nope, there is no way to end the guildleve early. The whole point of doing them is to gain you experience and gil/company seals (and later for items). That's exactly what the bonus is. For those five minutes, you should be killing as many of those creatures as you can. The experience from killing them isn't great, but what you'll get out of the bonus will make up for that handily, and is one of the more efficient methods of acquiring seals.

If you really don't want to kill more monsters, nothing says you can't just go do something else for the rest of that period. Once you complete it, you can always get teleported back to the guildleve quest giver.

Nope, I'm afraid there's no way to increase your item level beyond your level. Not while levelling, anyways. Every bit of equipment prior to level 50 has an item level that matches rather closely with its required level. Only once you hit max level does that open up into higher item levels.