They are roots. That type of squash roots all the vine. Since you are on a trellis, they can't help you much, but standard practice is to bury those roots wherever they touch the ground. I have had the base of a plant completely destroyed by SVB and the vine roots kept the plant producing. I am not sure what you can for an above ground wound. many times they will heal themselves. Perhaps a band aid as Dan suggested. A strip of plastic wrap like Saran does a nice job.

I'm not sure why that would happen, though I have not grown buttercup before. Buttercup is from the family c. maxima, which is more susceptable to borers than butternut. Butternut is in the c. moschata family. You mentioned purchasing plants. Why not get seeds during the winter and plant those? Squash is very easy to grow from seed. Just prepare your soil, so that it is weed free (or nearly so) and plant the seeds about 1" deep. It will take about a week for them to come up, but my guess is that squash planted from seed will grow as well or better than transplants, since there will be no transplant shock. Did you see pollinators on the flowers? I suspect you live in the Midwest. Some parts of the Midwest have been hard hit for pollinators. Below is a link to thread on hand pollination of squash. It's aimed at producing pure seed. But the technique would work for simply getting fruit. One could simply skip taping the flowers shut. George Tahlequah, OK Here is a link that might be useful: Hand pollination of Squash

It depends what you mean by "ripen". If you mean that the taste and texture improves, then maybe. But they will never be as good as those ripened fully on the vine. Further, the shells will not harden and become protective for long term storage, as noted. So plan to eat them relatively soon.

For the most part, the squash with the sunspots appear to be normal. There was one squash that had a sunspot which turned into a soft spot. I just cooked it on the grill and carved around the soft area. I can tell you that buttercup is a very tasty squash, and it has a very fine texture. I've brought in about three good sized squash per vine, so it would appear that the production is fairly decent as well. As for the corking/splitting on the stem, I view this as a sign that the plant and fruit no longer need each other. I've seen this with ripe watermelons as well. I didn't notice any frass that would indicate SVB.