What were the typical types of ships used in the Persian Navy in the 1730's and 1740's?

Short Answer

Details on the 1730s and 1740s Persian Navy can be pieced together to some extent from the Floor article you already found (but couldn't access), plus a smattering of other sources (including the Peter Good article found by Steve Bird). In short, prior to 1734 the Persians didn't have a navy of their own to speak of. From then until the late 1740s it grew substantially (although there was a major mutiny in 1740). It consisted of:

• frigate (as Steve Bird notes, these were at the time "relatively small, fast sailing ship[s]" which could be used for cargo transport or war)
• brigantine (two masts)
• sloop (single mast)
• gallivat, which " typically had one or two masts with a lateen sail, and some 20 benches for oarsmen, one on each side. They were generally under 70 tons (bm) in size, and had a prow much like that of a grab."
• ghurab or grab, to use the English term. For the Persian gulf, it is not clear exactly what these were but they had two or three masts and could vary significantly in size (described here as 'galleys').
• tranki (or trankey), single-mast coastal vessels which also had oars. The Persian navy had over 100 of these at one time and they most likely made up the bulk of the fleet numerically.

Details

Persian land operations had long been hindered by the inability to control the sea, with requests for loans of ships from the British and Dutch either ignored or resulting in only a couple of ships being sent for short periods of time. instead they sort of played a game of cat and mouse with Dutch and British trading companies, with the Persians asking to borrow ships and the Europeans, keeping in mind potential favours and penalties, attempting to avoid committing themselves but sometimes having to give in to Persian requests.

In 1734, the effective ruler Tahmāsp Qoli Khan (soon to become Nader Shah) began planning to rectify this weakness with his 'admiral' Mohammed Latif Khan. The latter acquired two brigantines from private English traders, but other details are sketchy and it seems that only one these brigantines saw action when Latif Khan's force of two ghrabs (lateen-sail rigged boats) and "40 assorted other vessels" was defeated in May 1735 when it attacked Turkish-controlled port of Basra.

In 1736, the Persians acquired several more ships, principally from the English; at least one of these was probably a 3-master. In October 1736, the Persian navy included two 400-ton frigates, 145 long and each with 20 guns. The bulk of the fleet, though, seems to have consisted of trankis (apparently not dissimilar to ghrabs in that they were "lateen-sail rigged boats" with planks sewn together with cords rather than nailed). Trankis may be the same as or similar to a terrada and were probably powered both by a sail and oars. They could be used for both cargo / transport and as war ships but were intended for coastal use.

Details for the fleet in 1742 are a little clearer:

Although Nader's shipbuilding plans failed to materialize, he nevertheless was able to get a considerable fleet by purchase, gift and seizure. By mid-1742 his fleet consisted of four 3-masters, three sloops, two gallivats, and a great many trankis, each with 4 to 6 cannon. Half of the four 3-masters had been purchased by Mohammad Taqi Khan for 7,000 toman from the English in Bandar Abbas. Each ship had 22 guns and was 110 feet long. The two other ships were at Bushire, where they had been bought from private English and French traders, each for 1,800 toman. These were 16-cannon ships, 90- 100 feet long. However, according to the Dutch, they were old, worn-out ships, from which the Iranians would not get much satisfaction. The ships of the Iranian fleet were mostly manned by Bengali sailors who deserted English and French ships calling at Iranian ports. Neither company could do anything about this, even when their sailors were pressed into Iranian service in front of their own factories. Moreover, Nader expected another eight ships from Surat, bought by Nizam al-Mulk. The fleet, which was well stocked with supplies, was to grow even more during 1742. Gradually the Hula mutineers were reduced to obedience, and their ships recovered, thus reinforcing Nader's fleet. Furthermore, the Imam of Masqat gave Nader two ships, one boasting 64 cannons.

Source: Willem Floor, 'The Iranian Navy in the Gulf during the Eighteenth Century'. In Iranian Studies Vol. 20, No. 1 (1987), pp. 31-53. The same text can be found in Kaushik Roy, 'Military Transition in Early Modern Asia, 1400-1750'

Peter Good also mentions a ship from Muscat, possibly one of the ones mentioned by Floor (but his use of 'gave' would need to be interpreted as 'was forced to give / hand over'):

In 1743, during the campaign on the Arab shore against Muscat, it is recorded that a ‘large ship from Muscatt of about nine hundred tons and mounts 50 guns arrived at Bandar Abbas after being handed over by the Muscati Arabs to the Persian forces.

Source: Peter Good, 'The East India Company and the foundation of Persian Naval Power in the Gulf under Nader Shah, 1734-47'. In A. Clulow & T. Mostert (eds). 'The Dutch and English East India Companies: Diplomacy, Trade and Violence in Early Modern Asia' (2018)

The Persians also had a small navy in the Caspian sea:

Nāder Shah had been encouraging the development of a small fleet on the Caspian in the 1740s. This was achieved by John Elton, an Englishman who had first arrived in the Caspian to open up trade with Persia through Russia. Once in Persia, Elton began building ships for Nāder Shah, and he had finished two frigates and four smaller vessels by 1745, with more under construction.

One of the vessels John Elton had built was "an 18-gun frigate". After this peak in Persian naval forces, achieved through purchasing, capturing and building, Nader Shah's navy declined due to neglect and shipwrecks, and his naval power never came close to matching that of his land forces. After his death in 1747, disputes led to the remaining ships being divided up among various commanders. The Encyclopædia Iranica article NAVY i. Nāder Shah and the Iranian Navy provides a neat summary of the problems which confronted Nader's maritime ambitions:

His naval project was up against a number of major obstacles unfamiliar to him. The lack of a naval tradition and the lack of expertise in the highly technical sphere of naval warfare was a problem, perhaps surmountable; more intractable was the established, self-sufficient character of the maritime Arab culture of the Persian Gulf region, on both the northern and southern littorals, which resisted outside interference. This manifested itself as a contributory factor in the problems of mutiny and piracy, but also in the way that conquests proved ephemeral and were quickly erased by new rebellions or invasions. Finally, there was an air of amateurishness and lack of seriousness about the enterprise. This was partly because Taqi Khan was involved. He was more an accountant than a commander. But elsewhere, especially if there was a persistent problem, Nāder went to the scene of the action and intervened personally. He never visited the Persian Gulf.