Q1: How does this card’s “Players can’t reduce play costs” effect work, exactly?

It prevents effects that reduce play costs. Even if you or your opponent activates an effect that reduces play costs, the cost that is paid cannot actually be reduced. Players must pay the full play cost printed on cards.

Q2: If this Digimon is in play, can I activate an effect that reads, “play <a card>...without paying its memory cost”?

Yes. Effects that play cards without paying their memory costs are skipping the costs entirely, not reducing them.

Q3: Can the play cost of a Digimon with the Royal Knight trait be reduced due to the Your Turn effect of my opponent's BT13-007 King Drasil_7D6 if this Digimon is in play?

No, they can't be reduced. This card's effect does not target Digimon, so even if your opponent's BT13-007 King Drasil_7D6 is in the breeding area, its effect to reduce play costs can't be activated.

Q4: When I have this Digimon, can my opponent no longer use BT14-046 Togemon's Your Turn effect or inherited effect to reduce the play cost of green Tamer cards?

That's correct. Such effects can no longer reduce play costs. This card's effect also prevents play cost-reducing effects from activating when Tamer cards are played.