Q1: I have 2 Digimon in this card's digivolution cards, and my opponent's Digimon digivolves with no digivolution cards. Does the digivolution cost increase by 2?

Yes, the inherited effect activates for each of the 2 Digimon, so the digivolution cost increases by a total of 2.

Q2: 2 of my opponent's Digimon with no digivolution cards DNA digivolves when my Digimon has this card in its digivolution cards. Does the digivolution cost increase by 2?

No, the digivolution cost only increases by 1. This card's inherited effect reads "When an opponent's Digimon with no digivolution cards would digivolve," so even in the case of a DNA digivolution where there are 2 of the target opponent's Digimon, the digivolution cost only increases by 1 with this effect because the digivolution only occurs once.

Q3: 1 of my opponent's Digimon with no digivolution cards and 1 of my opponent's Digimon with digivolution cards DNA digivolves when my Digimon has this card in its digivolution cards. Does the digivolution cost increase by 1?

Yes, the digivolution cost increases by 1.

Q4: My opponent's red Tamer digivolves into a BT4-011 Agunimon when my Digimon has this card in its digivolution cards. Does the digivolution cost increase by 1?

Yes, the digivolution cost increases by 1. BT4-011 Agunimon's effect reads "You may Digivolve this card from your hand onto one of your red Tamers as if the Tamer is a level 3 Digimon," so the Tamer is considered to be a Digimon without digivolution cards, and the digivolution cost increases by 1 due to this card's inherited effect.

Q5: My Digimon has this card in its digivolution cards. Does the digivolution cost increase by 1 when my opponent digivolves a red Tamer with cards under it into a BT4-011 Agunimon?

No, it doesn't. BT4-011 Agunimon's effect reads "You may Digivolve this card from your hand onto one of your red Tamers as if the Tamer is a level 3 Digimon," so a Tamer with cards under it is considered to be a Digimon with digivolution cards.

Q6: When a Digimon with no digivolution cards digivolves and the digivolution cost increases due to this card's inherited effect, what happens if I can't pay the digivolution cost as a result?

The digivolution fails, and the revealed card to digivolve into is returned to its owner's hand.