Don't worry about how you answered - that works. Sometimes I can't see new posts until I refresh the screen. I usually read all the posts, but maybe not everyone does. You can click the +quote then reply to a post, then another post, as many times as you like. Then answer all or each in one post. But you don't have to; what you did works too.
I've got the virginnia creeper, poison ivy, and sumac (but staghorn, not poisonous) all of which I've been trying to clear at least out of the fields and fence rows. Not the other plants you have (well, not in enough vigor to bother with yet); I also have several different invasive species too, though.
Since you already have the fence and posts, I'd talk to the neighbor who has the goats and see if he will pasture them inside your fence - with him taking care of them there. I'd run a water line out to fence (you will want it for your garden anyway). If you don't have a well in yet, then this after you put the well in. I would offer to pay him for the services of the goats because a half acre is probably not going to be worth him renting it as pasture for the goats, and he may need to feed them there or move them in and out.
I also have a neighbor with goats that I hoped would want to use some pasture. I haven't talked to her yet because getting the fence in hasn't reached the top of my priority list yet (partly because it is a 10 acre pasture and very overgrown). I did think about how it might work, though.
If your neighbor isn't interested, then I'd map out a perimeter (about the width of your lawn mower or twice the width of your lawn mower.) and keep that clear of any/all unwanted plant species. You should be able to do that (for many of the species you don't want) by just mowing if you mow often enough. Then as you have time and energy, work in from that perimeter - make the perimeter wider and/or make a path through the middle and/or make a clearing on one end. If you have enough time and energy then you will end up doing all of it. But if you try to do all of it and don't have enough time and/or energy then you end up bailing water in a sinking boat.
Along with the paths idea, and whatever else I did, I'd walk the paths frequently watching what plants grow and trying to learn to see the patterns of what grows how, when, and where. And walk through the middles occasionally to take out the tree of heaven sprouts and virginnia creep sprouts (with clippers and then paint the stems with BrushBGone or Round Up or something similar - look up what works best on that plant. The state wildlife service recommended this method as a good compromise between effectiveness and minimal chemical use. Like the paths, it is better to keep some areas completely clear and work on the borders of those areas than to try to do more of it than you can.
It is pretty late for planting wheat (uh, I don't know if that is a good choice in your climate, it might be). Anyway, it is not too late to plant buckwheat. Buckwheat is a good cover crop - both for green manure reasons and for crowding out undesirable plants reasons. You can plant it by broadcasting it by hand (5 acres took me ALL day; a half an acre should be doable)... anyway, take a handfull and throw it out in a sweep. Try to get it even but don't worry if it isn't. Plant it fairly thick. It will be more effective if you let the plants regrow (for ten days or so in my climate but it might be shorter in your warmer, wetter climate - your extension office or neighbors should know), then have them sprayed with round up and let that do its thing for a week or so. Then disc it under (optionally, but you will get a better stand of buckwheat if you do). Then plant the buckwheat.
Then, late this fall or next spring, see if you can hire someone to disc the buckwheat under or rototill it under. That should cost less than having the land cleared. Or you can rent a rototiller. That will help crowd out you undesirable plant species, help the field look nice. I saw you don't want a grassy field but maybe for a year while you prep it and wait for the right season to plant what you do want?
The landscape fabric under rock makes me shudder - I've had to deal with rocks as mulch in previous houses. The landscape fabric in a raised bed for your garden - great idea!
A colony of staghorn sumac is a clone - or all one plant- the "new" plants are coming up from the roots of the older plants as opposed to seedlings or such. So, it works best to take the whole colony out at once. August is the best time to do it (least vigorous regrowth). I'd do it by cutting each stem with lopers and painting the tops of the stems and down an inch or two with BrushBGone. Expect it to grow back - it takes about three years to eradicate it this way. The alternative is to pull enough of the roots out. My brother did that for a small colony (maybe 50'x100'); it is a LOT of work and he had a big tractor to do the pulling. I don't know how closely related poison sumac is to staghorn sumac.
Disinfect your lopers after cutting it.
I have a lot less trouble with virginnia creeper - just loping when I see the green growing in the dead vines of it is making good progress. But, I might be in marginal climate for it - so you might need to look up how to control it.
