There is a lot of discussion here, and if you can imagine a situation, it probably has already happened. Most allegations are real, but there are false allegations as well, and more often than not, the false allegations aren't of a sexual nature. Sometimes athletes, frustrated over a situation, hear something different than what you actually said. It happens between adults who work together, so we can expect it to happen with coaches and student athletes as well.
Be smart, and protect your student-athletes. Does that mean you cannot ever have a one on one conversation - no, but as has been said often, do it out in the open or in a situation where anyone can approach without any boundaries or restrictions. (Open doors, in sight of anyone passing by, an appropriate distance between you and student athlete, etc.)
If it involves discipline or an issue that you might not want to talk to an athlete on their own about, definitely involve another coach. I was at a spiritline competition at my school (I'm friends with the coaching staff), and was asked to listen in on a team discipline issue. I think these are the conversations that can be the most challenging and probably the ones that create the most tension.
If you're going to text, I prefer using an App that doesn't allow for deleting of text messages, so that no one can delete texts or be accused of that. And always keep it related to academics or athletics. But the key is that you text infrequently and communicate information relevant to your role. And most issues can wait until practice.
As for running with athletes, I usually had smaller teams, and sometimes I would run with a student-athlete individually - mostly when they didn't have anyone to train with, which occurs from time to time on smaller rosters. But almost all of our routes were out in public, and a number of them were out and back, so again, high exposure.
I can count the number of times I've ever given an athlete a ride in my vehicle. A few were during practice for safety concerns (weather and health) and had a coach with me. I know not everyone has had the luxury of an assistant coach or enough to do that. One I gave a ride to their theatre performance immediately after practice (assistant coach along with), and one I took home after a meet at midnight, after 90 minutes of calling the parent, the head track coach calling the AD and principal, and finally getting approval (parent was home passed out from drinking). I'd rather have not in all the situations, but sometimes things happen. Student athletes will find ways to get to practice via their peers, bikes, and even the deadbeat parents will figure it out.
Sadly, I've had peers accused of inappropriate activity (not many though). In one case, the accusation was false. The accuser wanted to recant even before the investigation. An investigation was done by the school district, another by local law enforcement. In both cases, he was cleared. No charges filed, the accuser apologized, and they went back to work. It would have been nice for the school district to have released a statement of support stronger than it did given they suspended him during the investigation. But the stigma will always be there. Now ... would I avoid coaching because of that? Absolutely not. I'm more concerned that parents will be upset that their athletes aren't competing in a meet, or I'm not using their private coach, or having mom and dad coach them during the season, and going to an AD and complaining and losing my job over that.
If your school doesn't have a strong code for coaches to follow, find resources like SafeSport and provide them to your site AD, and/or even your district AD. Be persistent. Talk to your peers about these things as well, and try to get them on board. There is safety and power in numbers. And you're doing it for the betterment of your student-athletes.
And finally, doing things the right way, and protecting yourself and student-athletes still provides a great chance to have strong coach athlete relationships , the kind that transforms life, and therefore persist far after high school. I've coached with former athletes, attended college graduations, attended weddings, visited with them and their newborn children, and been there when they lost family members. You can be smart and still develop the life changing, transformative coach-athlete relationships that help make athletics so impactful.