Q1: I used BT23-017 Betamon's inherited effect to play this card. Can I then use this card's On Play effect to DNA digivolve this card and another Digimon into a Shakkoumon in the hand?

No, you can't. BT23-017 Betamon's inherited effect can't be used to DNA digivolve a Digimon that can't digivolve.

Q2: I used BT23-101 Hudiemon with BT23-084 Erika Mishima in its digivolution cards to attack, and I used its 1st Alliance to suspend another of my Digimon. BT23-101 Hudiemon's When Attacking effect triggers at the same time, I use it to activate its On Play effect, play this card, and I use this card's On Play effect to DNA digivolve this card and the Digimon I suspended with Alliance into a Shakkoumon in the hand. Can I then use BT23-101 Hudiemon's 2nd Alliance to suspend Shakkoumon?

Yes, you can. If BT23-101 Hudiemon with BT23-084 Erika Mishima in its digivolution cards attacks, the 2 instances of Alliance and the When Attacking effect will trigger simultaneously. If you use the 1st instance of Alliance to suspend your Digimon, use the When Attacking effect to activate the On Play effect, and play this card, its On Play effect will derived trigger and activate first. If this card's On Play effect is used to DNA digivolve the card suspended by the 1st instance of Alliance and this card into a Shakkoumon, it will digivolve unsuspended, therefore the 2nd instance of Alliance can suspend that Shakkoumon. In addition, the DP that BT23-101 Hudiemon gains from the 1st instance of Alliance will remain, even if it DNA digivolves.