Steam: is it possible to use in-house streaming and record gameplay at the same time

As far as I know, you can't do that. If one of you wanted to play single player, then you could possibly set one computer to Offline Mode before starting the game, and then the other person should be able to log in on the other account and play multiplayer, but both of you being online at the same time should not be possible.

Keep in mind, this is technically not allowed. Steam's agreement says you're not allowed to share your account or your games, that includes your fiancee. I don't think they actually do anything about it as long as you're not talking about it on steam chat or their forums. However, you still risk losing your account and all your games when bypassing their rules.

EDIT: As of mid 2014, Steam has released Steam Family Sharing, which enables the sharing of Steam libraries. However, this system does not work on a per-title basis (only during the beta), but rather - If one title is being played, all other titles cannot be played (unless it's the same user that's using the library that is launching the second/whatever-is-after-first program).

So, this tutorial has a review of recording devices, including ones that support HD recording. It does note the following

I WILL BE LISTING SOME HD COMPATIBLE RECORDING DEVICES that enable you to record in standard and play in High Def simultaneously later in this article. However the prices can vary dramatically on these devices. There is only one somewhat affordable external device that allows true HD capture and that is the Hauppauge HD PVR and it is listed below.

It also says

As far as I know there are no affordable external devices that record 1080p or from an HDMI cable. The Hauppauge will do HD but it will only record in 720p or 1080i.

Mind you, this guide was posted 3 years ago so there is no guarantee that the devices haven't improved since then, but this seems to be a very complete guide.

Honestly, I think you're going to find better support on the GameDev side of things. A lot of players record their videos (especially those in MLG) but a lot of the times they won't care about audio syncing, or perfect framerates. You can take a look at this guide over at the MLG forums, but a lot of their focus is on cheap rather than quality.