Arixmethes can tap for mana the turn it enters the battlefield as long as it has slumber counters on it. If you can figure out how to untap it, that is.

Arixmethes isn't a land until after it has entered the battlefield. Effects such as that of Blood Moon won't affect its enters-the-battlefield ability. Similarly, replacement effects that modify how creatures you control enter the battlefield will see Arixmethes entering as a creature rather than a land. However, triggered abilities (such as landfall abilities) will see that a land has entered the battlefield and not a creature.

Arixmethes can't be played as a land.

While Arixmethes is a land, it's still green and blue, it's still legendary, and its mana value is still 4.

Once Arixmethes is a creature again, it can attack on the same turn as long as you've controlled it since your most recent turn began.

Arixmethes's effect causing it to be a land overwrites any earlier effects that gave it additional types. For example, a Phyrexian Metamorph that copies Arixmethes will be a land, not an artifact land, until its slumber counters are removed.

Arixmethes's triggered ability resolves before the spell that caused it to trigger, but after targets have been chosen for that spell. It resolves even if that spell is countered.