So far, I haven't had a tryout and then not invited the person to join us. I imagine it will happen. I was on the other end of this once; did the tryout, didn't get asked to join the team. I survived.
Most people I encounter seem cool and reasonable. So far, no problems on my team. I did have a problem with another captain this fall (she overreacted to a mistake and ripped someone a new one in front of the whole team), so I simply left the team without playing a single match. So yeah, there are certainly people out there who will make your tennis life miserable if you let them.
Eh, I'm competitive. It's a competitive sport. But I'm not so competitive that I'm willing to run the team in ways that aren't fair or reasonable or honest to gain an edge. I'm also not willing to hurt people's feelings so we can win. If we lose, we live with it.