Now days, I would recommend a USB connection. Fire wire can be a bit problematic if the FW chips are not 100% compatible. I've not had this issue but I hear that some folks have had problems.
Yes, the interface becomes the sound card for input and output. With the right interface, no mixer is needed. However, should you opt to use the mixer, that is totally fine. Yes, send it's output to the interface. Set levels accordingly. You can use the mixer for monitoring but I think you need to be careful because that could be a potential feedback loop. I recall this issue being discussed at length many years back by several here who were using the very setup you are describing.
My setup, as I mentioned, is a bit simpler. The interface is the common point and no mixer is used for anything going in.
I send the output of the interface directly to my studio monitors. The interface keeps the inputs totally separated from the outputs so no feedback can occur. I only have to be aware to turn the monitors off when the mics are hot.
My outputs echo my inputs and the playback so I have full duplex available at all times for sync recording. What I hear in the outputs is exactly what's happening in the tracks and the mix.
Have a listen to some of the music on my webpage that I have recorded using this simple setup. You might wish to consider using it as well.
When I first was getting started, I wanted to have a really "cool looking" studio..... you know, with the mixing desk full of faders and knobs and flashing lights. I was planning to use my D-1600 in that way. I had very little available desk space, and after setting it up, I was having a slightly distorted sound. That's when I decided to try the mic straight in and it was crystal. The D-1600 went on E-Bay the very next day.
I don't care what the studio looks like... to me, the important part is what sound quality I can get with the music. I've had a few singers come over to record some stuff, and I see a bit of disappointment in their eyes when they don't see what they are expecting to see..... but that changes when they hear what my humble studio can do.