What is known about this TSP variant?
This is a special case of the Travelling Salesman with Neighborhoods (TSPN) problem. In the general version, the neighborhoods need not all be the same.

A paper by Dumitrescu and Mitchell, Approximation algorithms for TSP with neighborhoods in the plane, addresses your question. They give a constant factor approximation algorithm for a slightly more general problem (case 1), and a PTAS when the neighborhoods are disjoint balls of the same size (case 2).

As a side comment, I think Mitchell has done a lot of work on geometric TSP variants, so you might want to look at his other papers.