We describe trust as a core value in our company and it goes both ways. We trust our team to do a great job, help each other when needed, be transparent, etc. They trust me and our execs to be open, honest, etc and I've gone above and beyond in that department. So I love the focus on trust in the 'running a remote team' section.
I'd also add: If you're planning on doing this as a founder/exec, expect to spend time and money dealing with different jurisdictions. Our team is mostly in the USA and for full-timers we have to file corporate taxes in every state where we have an employee. We have a team of accountants and HR folks that help us now, but it's still a lot of work and if it wasn't for my amazing co-founder, this would have been impossible to wade through.
For international domains, we have worked with or are working with people in Sweden, Greece, Paraguay and I think a few others. In general we reach out to our local law firm who connect us with counsel in those countries to figure out NDA's etc. It's not cheap.
Also had to work with local legislation in some countries that deal with workers comp type stuff and number of hours worked each week, government benefits - the list is long.
But the benefits are that we have an amazing, talented, diverse team who bring cultural influences from around the world and the USA and are able to live and work anywhere in the world they want to.
In terms of tools: VoiP is still ridiculously unreliable. I specifically include Skype and Slack in this criticism. As the team grew we discovered a secret weapon for super reliable voip calls: Teamspeak. It's AWESOME. It has push-to-talk which is great for big team meetings and you can see packet loss on a per user basis. You can also isolate yourself and your team to a single server so you're not load sharing with other teams during peak. It's been a real life saver.
Also, we don't do video. I know many remote teams do. We don't. What I love about it is that it really levels the playing field in terms of who is more charismatic on video. It also removes the distraction, frees up desktop space and reduces bandwidth usage for international team mates. When another company wants to do a hangout etc, we sometimes just tell them we don't do video and it's great. I can stand in my office staring out the window, really listening to what someone is saying, rather than having to think about whether I look good or watch them. Voice only for remote teams is awesome IMHO.
Thanks again to @WadeF and Zapier for doing this. I still find it absurd that remote work isn't the default for tech teams. One day we'll look back and wonder if perhaps lack of trust was the only reason big tech companies don't feel comfortable having everyone work remotely.