(Yugioh)What happens when someone stops these?

They do still activate regardless if the effect goes through.

Your 1st scenario I actually disagree with the ruling and think that the effect shouldn't go off. Why? Because in Yu-Gi-Oh a negated activation is considered never to have activated (as the game state sees it, thus why if Rescue Rabbit gets Divine Wrathed, you can activate a second one). So it flabbergasts me on why the discarded Atlanteans still retain their effects.

http://yugioh.wikia.com/wiki/Card_Rulings:Atlantea...

If "Atlantean Marksman" is sent to the Graveyard to activate a WATER monster's effect, and an effect like "Evolzar Dolkka" is chained to negate that WATER monster's effect, after the chain finish resolving "Atlantean Marksman"'s effect will still activate.[1]

The 2nd scenario I agree with, the effect should and will go through even if you use Veiler on Udine. The text says:

When this card is sent to the Graveyard to activate a WATER monster's effect:

So that means Udine activated its effect and got Veilered.

During your opponent's Main Phase: You can send this card from your hand to the Graveyard to target 1 face-up Effect monster your opponent controls; negate that target's effects until the End Phase.

Notice Veiler doesn't negate activations, only effects, thus why you can Veiler continuous effects since all Veiler does is negate the effect and not activation. The game state recognizes a negated effect that activated, as an effect that has activated (just the effects didn't go through). However a negated activations effect will have its effect deleted from the chain link and will be treated as never happened.

For discardng for Abyssmegalo, you still get the effects because the cards were discarded despite the fact that Megalo never made it to the field. If you summon Undine and send an atlantean, once it hits the grave, its effect resolves even if Undine was veilered because it hit the grave due to his effect (Even if it was later negated.)

I think you do, but I'm not 100% sure.