As for beta-ing, you've done it.

No known "right" way in the verse as far as I know. I did a log entry, but I've seen private messages, log posts, fic posts and engraved invitations.

I would be *VERY* careful of that sort of thing. As has been said before, definitely contact the original author - the last thing you want is to end up on some list of plagiarists, which I've seen happen.

Unless you get implicit agreement and blessing from the author, I just wouldn't do it.

Just my $0.02

I agree with SigmaNunki - you would really need to try and contact the writer first. If you go to their profile you'll be able to see when they last visited, so that would give you an idea if they're honestly no longer a Browncoat (a terrible thought, but I suppose it must happen once in a while) or if it looks like they're still lurking. Send them a PM, and ask politely.

If you don't get an answer after a month or two, I'd say post a chapter, but making absolutely sure you cite the original writer, and explain at the top of the fic. If they then complain, you can still take it down.

Whatever happens, keep writing.

I'd say that you should definitely talk with the person of the fic *before* you do anything. But, in general, people can rarely take over such things. The general tone and whatnot is typically unique to a particular writer. So, it's probably not a good idea.

Why don't you just start your own fic?