Nethergoyf's first ability counts card types, not cards. If the only card in your graveyard is a single artifact creature card, Nethergoyf will be a 2/3. If your graveyard consists of ten artifact cards and ten creature cards, Nethergoyf will still be a 2/3.

If a card with escape is put into your graveyard during your turn, you'll be able to cast it right away if it's legal to do so, before an opponent can take any actions.

If a card has multiple abilities giving you permission to cast it, such as two escape abilities or an escape ability and a flashback ability, you choose which one to apply. The others have no effect.

Card types that can appear on cards in a graveyard are artifact, battle, creature, enchantment, instant, kindred, land, planeswalker, and sorcery. Legendary, basic, and snow are supertypes, not card types; Lhurgoyf, Forest, and Siege are subtypes, not card types.

Escape's permission doesn't change when you may cast the spell from your graveyard.

The ability that defines Nethergoyf's power and toughness works in all zones, not just the battlefield.

After an escaped spell resolves, it returns to its owner's graveyard if it's not a permanent spell. If it is a permanent spell, it enters the battlefield and will return to its owner's graveyard if it dies later. It can escape again.

If you cast a spell with its escape permission, you can't choose to apply any other alternative costs or to cast it without paying its mana cost. If it has any additional costs, you must pay those.

To determine the total cost of a spell, start with the mana cost or alternative cost you're paying (such as an escape cost), add any cost increases, then apply any cost reductions. The mana value of the spell remains unchanged, no matter what the total cost to cast it was and no matter whether an alternative cost was paid.

Once you begin casting a spell with escape, it immediately moves to the stack. Players can't take any other actions until you're done casting the spell.