I'll share what works for me, and you decide whether or not it is appropriate to your situation. I have had great success with composting in place / hybrid lasagna gardening to keep the grass and weeds out of my gardens. I have a few raised beds in my lawn and I keep a border which I think sounds similar to what you have or are wanting to do. I keep it simple and mix my freshly cut lawn clippings with used coffee grounds and a good amount of cross-cut shredded newspaper. I put this mixture down generously (in inches deep) over the area where I don't want any grass/weeds coming up and then I top it all off with loads of hay/straw (I don't know which is what, but it comes in a bale so I will call it "hay") to keep the sunlight out. My thought is that under the layer of hay, the fresh clippings, UCG and newspaper eventually heat up and kill or severely weaken the grass and weeds. The hay seems to prevent any additional germination by keeping out the sunlight. Add more hay as necessary to keep area covered. The sun does a nice job of bleaching the hay in place, so I think it is an attractive mulch that does a great job. I used to think this was my idea since I just took a chance on it out of frustration of battling (and losing to) the weeds and "lawn creep". But, it's not a new idea. Here are a couple of links that I discovered after the fact. Wish I had read these a long time ago... Ruth Stout And Permanent Hay Mulch Hay Makes a Great Mulch

I agree that is dodder. I've just seen it first hand once, I found it in a clump of Echinacea/cone flower in a long perennial bed I shared with a neighbor between our homes. After ID, I hand picked every single thread of it and put it in the garbage, kept a very close eye on the area watching for recurrence (hands and knees) the rest of that season and again the following spring. It never returned.

2nd photo is aphids - I've observed lots of them on new growth of my grapevines as well. You can wash them off w/ a sharp spray from a hose. I leave them alone & let the ladybugs take care of them - they never seemed to affect the flowers or fruit & vines eventually outgrew them. 1st may be a female cottony cushion scale insect - I just rub them off the plants w/ gloved fingers. Both are sucking insects

I'm guessing it is some type of fountain grass. As long as your neighbor has it you'll have a ongoing problem. The seeds from their grass will be blown into your lawn causing it to regrow (unless the seed heads mysteriously disappear). I agree with the roundup just wait till it starts growing. You can spot spray it to minimize damage to your lawn.