If you got paid the week of 'connectivity issues' and did not work, then he has a point.
On the general issues - verbal contracts are always dicey. They do hold weight, depending on the state, the terms, etc. I would get something in writing, if nothing less at least get an email indicating the intent and general terms of your employment. However, most companies can terminate at will, and contracts generally only bind them to causes of action and severance packages (Unless you're an NEA teacher).
No. I billed for one hour that week - the time I spent in the office trying to make things work after the long drive down. I do not - nor expect to - get paid for time not worked.