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There are several products on the market that claim to kill grass, not plants, but from experience, I can tell you don't get carried away with the stuff. The good news is that the only thing tougher than your weeds is your monkey grass. Each of those individual clumps is a seperate plant, so while you may damage one, the others will survive. You can't mass spray them, but it does give you a little leeway when you get up close to squirt. You can use an all purpose vegetation killer. I cut the bottom off a small coffee can and set it over the weed to block the weed killer from hitting the plants I want to save.

You can use monkey grass to border the vegetable garden, but you might want to consider flowering plants that attract beneficial insects and those that repel bad bugs, like marigold and nasturtium. Here is more on monkey grass:
https://www.gardeningknowhow.com/ornamental/groundcover/monkey-grass/monkey-grass-a-spectacular-groundcover-for-the-lawn.htm

Yes, it sounds like this is caused by too much water. We would recommend that you cut back on the watering to correct the yellowing problem.

Yes, they can be used to collect seeds once the berries (fruit) have ripened. You can then collect, clean and dry the seeds. Be sure to remove the fleshy coating on seeds before storing. One thing to keep in mind, however, is the fact that it will take much longer to grow monkey grass this way. It is easier and faster to simply divide the plant.

Here is more information on growing this grass from seed: https://www.gardeningknowhow.com/ornamental/groundcover/monkey-grass/monkey-grass-a-spectacular-groundcover-for-the-lawn.htm

Monkey grass and hummingbird vine (aka: trumpet vine) are generally hardy in Zones 5-10. Both should do fine in your state with exception to the most northern parts of New York, which are Zones 3/4 and will require winter protection.

While I would recommend transplanting in fall or even spring, as long as you provide them with adequate water while they are establishing themselves, you can do so now. It also helps to make certain that all the roots are included when lifting and sometimes pruning them back will help alleviate transplant shock as well.