First off, congrats. What a property. I'm a minimalist for country properties - the more you can leave it to nature, the less work for you. I think you should take bites out of this so you don't get overwhelmed (unless you are just looking for ideas and plan to hire a contractor).
I like Bondo's idea of a rock swale. They have even become trendy landscaping features for people who don't even have a drainage issue (dry river bed). I would use no larger than fist-sized river stones and lay down filter cloth. If there is a flatter area along the course, yes, let it widen out a see what grows (no cloth). If you get enough rainfall, it may turn into a bit of a wetland.
The brush, yes, whack it out, burn it or haul it away, rip out the roots, plant grass or whatever you want. Depending on what it is, eradicating the roots permanently at once may or may be successful. Roundup usually doesn't work on anything that is that woody.
Around the house - yup, it is past its prime. If you tried to cut back those bushes (they look to me like cedar or juniper, but whatever) I think you'd end up with a big stump. At some point you simply have to say its time. Find some bushes you like that maybe flower at different times of the year, and make sure to keep them off the foundation - they grow.
Your slope - I can certainly see how it would be a pain to mow. Even if you put in a retaining wall, unless you made it really high it looks like you'd still have quite a slope. Mulch maybe?
There are grasses that will grow under trees but it is always a challenge. As well, it looks like a lot of the soil has become compacted. You might have to aerate and top dress if you want grass. You might want to consider ground cover like floxx or periwinkle. Remember, the more grass the more you have to mow.