I have seen your weather this time of year. It can be dreary at times.

I'm not sure what your end goal is here, and you have a pretty big lawn. The best way to keep things orderly that I can think of is to get some sheep. You can get some feeder lambs, let them keep everything down until the weather dries out, then sell them at a heavier weight and hopefully get your money back plus more. Since you have corrals, I would assume you have some livestock. But, if the livestock are leaving some types of plants, then mix in another animal species. Both horses and cows have their forage preferences. If you have either of these, mix in some sheep and they are likely to eat what the cows or horses won't. Temporary electric netting will allow you to graze sheep wherever you want for short periods if fences aren't available.

From experience, mowing wet grass can plug up the mower, requiring frequent stops to clean out the mower platform. It also tends to shred the grass rather than cutting it, which leaves a rough appearance and creates more opportunity for diseases to enter the plants. You can still mow when wet, your results just aren't very good, and you might leave ruts in the ground that will show for a very long time.

Herbicides may or may not work well at your current temperatures, but you will need to know what you are trying to achieve. Roundup will kill everything and you'll have 20 acres of bare ground. Bad idea. A broadleaf herbicide such as 2,4-D will control most broadleaves and leave the grass untouched, or a grass killer will take out most of the grasses and leave the broadleaves, both of which you will still need to mow. You can apply a half rate of a grass herbicide such as Poast or Select and it will make grasses sick and stunt their growth without killing them. I'd stay away from herbicides except for specific problems or small areas, though, and go with livestock, and wait for some dry periods to mow close around buildings and a front lawn on the house.

Wet is relative. Really wet grass might clog both reel and rotary mowers but even the commercial mowers are out there cutting grass when it is damp with no problem. Grass wet with dew can be mowed, although it will tend to clump more then dry grass would, but I've not had it clog any mower I've used. If you get no rain for a day you could mow in the evening and should not have a problem.
To a certain extent mowing before any "weeds" set seed is the best way to keep them controlled. Keep in mind that the highest court in France found that the manufacturers of some of the glyphosate products lied about how environmentally friendly they are and supported the ban on sale of those products.