Does a thunderstorm or any other storm other than Tropical storms and Tornadoes have an Eye?

Question

My grandfather and I were wondering this when our power went out and back on, after that, there was no wind, just thunder for about 5 minutes, then, it started right back up again.

Answer 1

Not really, though some of the powerful non-tropical lows can have a relatively calm center.  What happened in your case there is that there were a few storms that moved along the same general track, and a short time after one storm moved away from you, the next one moved by.  This is not an unusual event.

Answer 2

Generally no, but sometimes it may give that appearance, as a Vorticity maximum wraps around itself creating a circular appearance, but generally no eye.

What you were experiencing is most likely the lull associated with the area between 2 thunderstorms, or a break inbetween 2 lines of showers/t-storms.