If there's no equipped creature as Skyclave Pick-Axe's landfall ability resolves, no creature gets +2/+2.

Attaching an Equipment with its enters-the-battlefield triggered ability isn't the same as using its equip ability. You don't pay mana for the attachment, and if the Equipment enters at a time you couldn't cast a sorcery, you can still attach it to a creature you control.

The bonus from Skyclave Pick-Axe's landfall ability applies to the creature it's attached to at the time the ability resolves. That bonus doesn't apply to a new creature if Skyclave Pick-Axe becomes attached to a new creature.

If the target creature becomes an illegal target, the Equipment remains on the battlefield unattached.

Whenever a land you control enters, each landfall ability of the permanents you control will trigger. You can put them   on the stack in any order. The last ability you put on the stack will be the first one to resolve (As a result, you can have those abilities resolve in the order of your choosing.).

A landfall ability doesn't trigger if a permanent already on the battlefield becomes a land.

A landfall ability triggers whenever a land you control enters for any reason. It triggers whenever you play a land, as well as whenever a spell or ability puts a land onto the battlefield under your control.