Yes, you get a ring under those conditions. After you make the bonds, there’s no way for the system to know the difference between a bond joining the ends and a bond somewhere in the middle. The numbering convention is just for bookkeeping purposes; it has no physical meaning. Therefore, if your system is large enough that no atoms interact with their own images and no bias exists in the initial setup, the ring aspect won’t matter for most purposes. Feedback may be a problem for certain properties if your system is too small, but I don’t know of any other way to get an infinite system that needs to be continuously connected.
LAMMPS does calculate the minimum image on bonded interactions if PBC are specified. Think about what happens to a dimer that is halfway across the boundary. Technically, a separation distance that is nearly a whole box length could be computed, but the minimum image ensures that the calculated bond length is the one you expect. For the ring example, it’s unlikely that the bonds between first and last atoms will remain exactly at the boundary. When the fluctuations cause the bond to be entirely in one box, why would you expect the bond to register as being nearly a whole box length? The PBC is just a convenient reference point. There’s no wall or other actual boundary in space. A bond across a boundary is no different than a bond anywhere else. Bonds don’t change merely because a spatial reference point has been crossed.
For my recent simulations of graphite in infinite sheets, I had no problems with bonds connecting one end of the sheet to the other end in two directions. Thus, I expect there would not be a problem with nanotubes, although I have never done the simulations.