I only read the manga and haven't finished the OVA, but as long as they keep following the same storyline here's the answer.
Did you notice that Schrödinger was anywhere instantly, including in peoples imaginations? He is simutaneously everywhere and nowhere, which is a fucked up concept I know and hard to understand. They compare him to the Cheshire Cat or an imaginary number. He is named after the Schrödinger's cat thought experiment, which is appropriate due to his ability. He is imortal and omnipresent. If you're better with philosopy he is a manifestation of "I think, therefore I am." As long as he thinks he's alive, he is. As long as he wants to be somewhere, he is; which makes him a great spy for the Millenium.
It's like he has one foot in existing, and the other in nonexisting. He's part real and and part imaginary. It's a difficult concept to get, but this concept is also a very important quantum physics concept and a lot of scientists in the field have a hard enough time wraping their head around it.
Alucard's main ability is to absorb others to become a part of him including any abilities they might have. That is how he can summon an army due to having absorbed so many people, and drawing on the vampires' powers he had absorbed earlier in the series. The whole point of defeating Alucard, was for him to absorb Schrödinger. Since there were so much bodies he tried to absorb at once so he didn't realize he also absorbed Schrödinger in the process. Since it's hard to control his powers due to the fact Schrödinger's powers teters on the brim of existence, Alucard simply ceases to exist and is defeated.
In the end (of the manga at least), Alucard returns to Indegra having mastered Schrödinger's ability. So now he once again exists and now is omnipresent as well.
The whole Waler thing wasn't explained well since even in the manga it is unclear where he first joined the Millennium and betrayed Hellsing. He was either brainwashed by the Doctor, or was with them since his childhood. The anime OVA is based on the manga, and since the manga's auther is said to write a prequal that may or may not clearify this confusing event. The animators I guess chose no be ambivalent in the anime as well.
Anyways the Doctor, (the guy with the many lens glasses, parted blond hair) has the ability to make artificial vampires, and created the werewolves. He either made Walter into an artificial vampire or just deaged him. Apparantly the deaging was unstable and changed him back to a child.