So, around this time most of the English navy was still comprised of a mixed bag of former merchant ships and some private warships, with a sprinkling of "King's ships" (or, rather, Cromwell's ships) among them.

The State's Navy (renamed as such under the Protectorate) in the 1650s started to build longer, lower, faster ships to combat the cruisers from Dunkirk that were preying on English trade. These ships began to be called "frigates" and were distinguished from earlier ships by having a larger length-to-width ratio than earlier building, and by having low or no forecastles and cut-down sterncastles. The Constant Warwick of 1645 is thought to be the first custom-build of this type; she was privately built but chartered then bought by the State's Navy.

The Dunkirker ships were long, low and relatively small, designed to prey on shipping across the channel, and did not have much depth of hold, meaning they were ideal for quick raiding activities but couldn't keep to sea for very long. So they were not ideal for the types of missions that had to be carried out by the State's Navy, and so the "frigate" type ship became larger fairly quickly. By the 1650s, the navy built the Speaker as a two-deck ship carrying 44 guns, later increased to 62 after a rebuild in 1688. The Speaker is the ancestor of the multi-deck English ship through at least the 1750s or so. But these "frigates" were always part of a mixed fleet that would include ships of an earlier design, or merchant ships.

This is a long way of saying that if you want your character to be on board a ship of whatever type, you can plausibly invent a justification for it -- just don't go for more than 50 or so guns and you'll be within range. The National Maritime Museum at Greenwich has tons of ship plans and models online: https://collections.rmg.co.uk/collections.html#!cbrowse

I wrote about all this stuff a bunch more here.

Moving on to your question about storms -- that older thread isn't bad. It really does depend on the type of storm and type of ship that the captain has to deal with that would inform his tactics for fighting the storm. In general, if they had warning of, say, a hurricane or other storm, the first thing to do would be try to win as much sea-room as possible from whatever lee shore they were near, accompanied by striking down as much of the top-hamper as possible. If they were reasonably far at sea, they could scud in front of the wind with minimal sail at the foremast, to keep the bow pointed away from the wind. Ships in 1655 were still steered by a tiller connected to a whipstaff, with the helmsman standing above the tiller flat and looking out of a port -- in case of a storm, they could have extra men in that space but it would be cramped and difficult. Ship's wheels don't come in until the 1690s or so.