Can I talk about articles I'm refereeing?
There are different confidentiality requirements involved, some to protect the authors, some to protect the referees.
A) As a referee, you absolutely must not reveal the content of an article you are refereeing to others. In many cases, the paper will be available on the arXiv anyway, and you can speak about the contents of the arXiv preprint, even if you are refereeing the paper, but not about anything you couldn't have learned outside of the refereeing process.
R) It is not customary to reveal oneself as a referee of a specific paper, but there is no strong consensus on how strict this should be handled. It is quite certainly ok to mention that one is refereeing for a specific journal (many people list the journals they have refereed for in their CV).
To your detailed questions:
When discussing the article, do not mention that you are refereeing it. Be careful not to mention details exclusive to the submitted version and absent from the preprint.
When ask by your office mate, it is fine to say that you are refereeing. They should not pry for details. If they know which article you are refereeing, do not mention the journal. If not, you can mention the journal. Better not name the editor.
As mentioned in the comments, it is typically better if referee reports are independent. Do not discuss an ongoing referee report with your co-referees.
Some people do reveal themselves as referees to authors. In general, I would advise against it though.
First, it's important to distinguish two cases: The case where the work under review is already publicly available as a preprint (which seems to apply here), and the case where it isn't. In the latter case, the work under review needs to be handled with absolute confidentiality. In the former case, things are less clear.
I will answer your questions for the case I feel confident about: a non-publicly-available submission in software engineering. For a publicly available submission, adequate answers may be less (but not more) conservative.
Is it ok to say that I'm refereeing the article? [...] Can I bring up my task as a referee? If I should hide it, what are some recommendations to do it?
It's OK to say you're reviewing "an article" or "the article on my desk". It's not OK to disclose the title or author name.
Can I name the journal or the editor [...]?
Yes. The information that someone has been reviewing for a specific journal is routinely mentioned in CVs, and the list of editors of a specific journal is normally publicly available, too.
I also happen to know another referee of the article. Is it appropriate to bring the review process up in a conversation?
Yes, but only after both of you have submitted their reviews, and if the conversation involves no additional persons other than the referee.
And the killer aspect: If I meet the author on a conference, is it acceptable to tell them that I was a referee?
I've occassionally seen cases where reviewers revealed their identities even within their review, which effectively amounts to the same situation as in your question. However, in fact, this seems to be a controversial question without a universally accepted answer.
My answer is going to differ from the comments currently left.
Ethically, there's a certain reserve you need to have around the articles you are reviewing. You certainly can't share the manuscript.
But what you are talking about is coffee break conversation. I see no issue in saying that you are reviewing an interesting manuscript, on subject X. You should not revealed the authors' names, but you can certainly talk about it. Reviewing is not a secret mission.
You should not however discuss it with the other referee, as you don't want to influence each other.
As for revealing your identity to the authors, I personally see no problem with it, especially if you had a positive review. Be careful, however, as if their article got rejected, they might not like you very much...