How many storms at sea do sailing ships typically encounter on a voyage across the North Atlantic?

Typically at least 1+. The North Atlantic is a convergence zone for climate/weather, it was then/ is now a stormy area.

In the Age of Sail sailing from America to Europe (W to E) you had to go that route as that's how the wind and currents went. From Europe to America (E to W) you went a more southerly route through the N mid Atlantic (again for wind and current) and had fewer storms but faced the doldrums (wind dead zone) of the 'Sargasso Sea' well off east of the Carolina's lengthening your trip.

From America to Europe in the Age of Sail figure 25 days or 1 month for the trip sure to have 1 storm/ probably 2, and some trips as many as 7. 

Time of year course mattered too....most storms America to Europe Fall/early winter...most storms Europe to America late summer Hurricane season.

The number of storms is largely dependent on the time of year 

a) Depends on the time of year. 

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b) Weather patterns fluctuate over time. We don't actually know much about what the weather was like in the North Atlantic 500 years ago because there were few ocean crossings and therefore are few records. This illustrates how few major storms we know about: https://en.wikipedia.org/wiki/List_of_European_win... When speaking of the past you need to change the word tenses.

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c) Don't confuse fictional movie melodrama with reality.The chance of a lightning strike setting sails on fire is quite low, but it did happen. In 1852, British inventor Sir William Snow Harris published the first systematic study of lightning strikes on wooden ships. He collected data from 235 strikes on British navy vessels from 1793 to 1839. The damage typically consisted of “shivering” or splintering of the mainmast: Long shards of wood flew in every direction, sometimes wounding a sailor or knocking him off the deck. Sails and rigging might catch fire, requiring officers and crew to smother the flames with the aid of the rain and wind. None of the ships in Harris’ sample was recorded as being obliterated, and the vast majority were repaired by their crews and continued sailing.

There is no set number. Sailing west across the Atlantic begins in November (the end of the hurricane season), if you were lucky you might encounter no storms, if you were unlucky…

Lightning strikes on ships is possible but by far the majority miss the ship, the height of masts can make it more likely that lightning strikes.

St Elmo's Fire was well-known and is sometimes an indicator of lightning.