No. As per Necrovalley's text, no card can be banished from the graveyard while Necrovalley is in play, there are no exceptions.
Necrovalley's second effect that prevents cards going from the graveyard to other zones, apart from by their own effects, does not override this. It would let a card with special summoning conditions summon itself, or allow a card to return to hand, if they are that card's own effects. But a card that has to banish itself will be prohibited from activating due to the first part of Necrovalley's effect.
Necrovalley does the following, "Cards in either player's graveyards can't be banished" "Negate any card effect that would move a card in the Graveyard, other than itself, to a different place" The only way this affects the Dragon Ruler's special summon effects is it will prevent monsters from being banished from the graveyard as the cost. They will have to come from hand instead. But as Necrovalley doesn't stop a card moving itself out of the graveyard, the effect will not be negated.
In general, a monster's effect can only be used while it is face-up on the field. If it can activate in a different zone, such as hand (Gorz, Emissary of Darkness) graveyard (Necro Gardna) or while banished (Stardust Dragon) then it will say so on the card. This is why a card such as Cyber Dragon cannot freely summon itself from the graveyard every time the condition is correct - it summons from hand. Evil HERO Infernal Gainer has no such text, therefore the cost of banishing it to activate its effect, can only be performed while it is face-up on the field.
Colossal Fighter's effect triggers after he hits the graveyard. That means that he himself is in the graveyard when he activates, and he is a valid target for his own effect. So yes, he may summon himself.
No, Aqua Spirit can only be special summoned from hand that way. The only time a monster can summon itself from a different zone, is if it specifically says it can. Plaguespreader Zombie is one - it states that it summons itself from the graveyard. But without such text, then they can only summon themselves from hand.
No, unless specifically stated otherwise on the card, a monster's special summon requires it to be summoned from your hand, or in the case of Synchro and XYZ monsters, from the Extra Deck.Plaguespreader Zombie can special summon itself from the graveyard because it says it can. But you cannot use for example, the special summon requirements on Chaos Sorcerer, or an XYZ monster, while they are in the graveyard.
Necrovalley does the following, "Cards in either player's graveyards can't be banished" "Negate any card effect that would move a card in the Graveyard, other than itself, to a different place" The only way this affects the Dragon Ruler's special summon effects is it will prevent monsters from being banished from the graveyard as the cost. They will have to come from hand instead. But as Necrovalley doesn't stop a card moving itself out of the graveyard, the effect will not be negated.
Necrovalley will be activated, and when it resolves, Karakuri Showdown Castle will be destroyed at the same time as Necrovalley's effects becoming active. Due to the new rulings, Necrovalley doesn't necessarily care about cards in the graveyard being targeted, but monsters in the graveyard can only be special summoned by their own effects. So Treeborn Frog can special summon itself, and Colossal Fighter is allowed to target and summon itself, but not other Warriors. Since the Lv4 Karakuri monster in this case is not summoning 'itself', and it is another card effect trying to do so, then it will not be able to be special summoned due to Necrovalley's effect.
Unfortenately it can only connect to itself not other WC Games
No. The effect of any monster face-up on the field is negated once sent to the graveyard, unless the card itself states that the effect still works while in the graveyard
In general, a monster's effect can only be used while it is face-up on the field. If it can activate in a different zone, such as hand (Gorz, Emissary of Darkness) graveyard (Necro Gardna) or while banished (Stardust Dragon) then it will say so on the card. This is why a card such as Cyber Dragon cannot freely summon itself from the graveyard every time the condition is correct - it summons from hand. Evil HERO Infernal Gainer has no such text, therefore the cost of banishing it to activate its effect, can only be performed while it is face-up on the field.
Yes, it can, as it itself is an 'Iron Chain' monster, so can send itself as its own cost (note that it is not a 'tribute').
If it is originally in your hand, then yes, you may send the Machina Fortress in hand as well as one other monster, and then special summon that same Machina Fortress from the graveyard.
Yes. Once Legendary Six Samurai Shi En is successfully summoned to the field, then it can use its quick effect against spells or traps the opponent activates, even one activated in response to its own summon.
No, unless stated otherwise, a monster special summon effect works from the hand. Treeborn Frog special summons itself from the graveyard, because it says so. DD Survivor special summons itself from the removed from play zone, etc. But Dark Armed Dragon can only special summon itself from hand.
Colossal Fighter's effect triggers after he hits the graveyard. That means that he himself is in the graveyard when he activates, and he is a valid target for his own effect. So yes, he may summon himself.
If an equipped monster finds itself no longer equipped to a monster for any reason, then it is sent to the graveyard.
Yes. The first Inzektor can use its effect to equip Inzektor Hornet to itself, Inzektor Hornet can then send itself back to the graveyard to destroy one target. The second Inzektor can then repeat the process.