Re: What version to use for a new project?
I tested both.
1.0-alpha integrates better but is more unstable and there are lots of errors. I would no use it on a live site.
0.9.0.4 is much stable, and runs fine. I used this plugin https://bbpress.org/plugins/topic/freshly-baked-cookies/ and then forced always to login/logout from WordPress. That works perfect. Login/out from bbPress does not run correctly in this situation, so I preferred to force to login/logout from wordpress.
In my case I was talking for a project 6-9 months vista, to be finished later this year. This makes things a bit different as I hope then bbPress 1.0 will be more mature.
Also consider that you always can upgrade from 0.9 to 1.0… but you can NOT to downgrade from 1.0 to 0.9. So, perhaps I would start from the stable version.