I'd say there even some advantages as you get to meet more people, get more ideas and generally your brain works better while on the road. (My theory for that is when we are in a foreign environment we need to be more careful)

I run an online business and live two kinds of life: 1) Travelling or staying a couple of months in each place. 2) Home, Italy, where I spend most of the time at my desk.

I was actually able start it (no funding) and run it while constantly on the road.

Define "people". I could see your bank wanting you to have a defined location so they can send you snail mail. Your customers don't really care. Take steam or amazon for instance. As long as I can login and play games or buy books, I don't care where they are or if where they are changes every week. Customer service is certainly easier if you are in the right timezone, but not impossible from others.

Being in a stable location is always useful - easier to contact, and makes it possible to have face-to-face meetings. However, a startup is about the best (tech) job for travelling (on your own terms). I imagine people fly out for meetings, doesn't matter where they're flying from.

Hasn't been a problem. We have a mail service and US phone numbers, and so most of the people we have to formally do business with assume we're in the USA.

There is the impact of pulling up roots and moving to another country... and when you're working really hard there's a lot less sightseeing than there would be otherwise.