At what stage to the marines go to the Deathwatch? The text in the book would infer that they were veterans, but I could not find anything say this was true.

To my knowledge (don't expect references, it is getting late here and I'm just throwing this out there) Chapters sign an accord, either at the original Conclave mentioned in the Core book, or later in their history, to provide warriors to the Deathwatch. There is no specific limitation, but Deathwatch are a 'specialist fighting force' and so it is to be expected that there are vastly different selection criteria between chapters, reliant on valour, knowledge and wisdom in fighting the Xenos. even the most dense DW battle brother is likely a bit more book-smart concerning alien-killing than the 'average' (if there is such a thing) Marine.

In my humble setting, the criteria differ between Chapters, but core concepts are; Heavy combat experience against multiple Xenos threats. Honours amongst their own for such actions. Are not integral (yet) to the running of their original formation (at least in their first term of service).

It is up to you, if there is no official text, what the standards are, but I do believe that they should have been picked out for exemplary and skillful action, and/or the potential to absorb vast quantities of information from their service in the Deathwatch. It isn't just about sending their best to DW, but getting them back better than ever.

I read in the Deathwatch Core book that they start as Scout Marines, then on to the various stages of Devastator, Assault, and Veteran squads. I am assuming that the Deathwatch candidates have to be pulled from the Veteran company of each chapter. They have the same requirements. Or at least they have to qualify for the Veteran company. I could see it possible that instead of transferring to the Veteran company, a Space Marine is sent to join the Deathwatch.

That sound right to you guys?

As I think its been mentioned it all comes down to each Chapter. Some may even send members to the Deatchwatch as a form of punishment. I of course don't think that its the members from the veteran company that are sent but brothers who have shown themselves reliable and skilled in battle and can be dependent on to not shame the Chapter.

My understanding is that any marine can be seconded to the Deathwatch, the requirements vary from chapter to chapter but on the whole they have to prevail against numerous Xenos threats. But they could also be sent because a chapter doesn't have experience fighting a particular foe and seconding a marine to the Deathwatch is the only way to get the information before they deploy chapter companies against the threat. It could also be because they displayed an exceptional skill of arms and seconding them to the Deathwatch will allow them to become even deadlier.

As people have said marines can also be seconded as a punishment for something they did in their home chapter that warrants them begin exiled for a period of time.

Page 309 - 310 in the core rulebook describes the Deathwatch's recruitment methods.