It's not really that easy to answer many of the questions you posted without doing some original research. But, I dug up my notes to see what I had on Graves. Also, the A memorial: or, a short account of the Bahama­ Islands is a short little 7-age publication from 1707 that probably isn't available unless you go to a college and access the Eighteenth Century Collections online (a database full of pdfs of eighteenth-century English publications).

Anyway, what notes I do have on him are below:

In December of 1686, a sloop with colonists from Jamaica went to the Bahamas, and one of their number was John Graves.

In 1692, John Graves helped a Bahama governor, Cadwallader Jones, bring a political opponent (who had Jones briefly jailed) to trial by sailing to England and obtaining a commission to try his opponent in the Bahamas. Jones was known as a corrupt Governor (kind of par for the course for the Bahamas for this period), which is why some attempted to lock him up in the first place.

In 1697, Graves was made the Collector of Customs.

In 1701, a group of men that included Read Elding (a former lieutenant governor of the Bahamas), John Warren (the Speaker of the assembly), John Graves (the collector of customs), and Ellis Lightwood all opposed the current governor, Elias Haskett, and (at that time) Vice-Admiralty Judge Thomas Walker. The latter two engaged in various illegal and self-serving activities. So, the group of Bahama citizens I mentioned conronted Haskett and Walker. Haskett had Elding imprisoned for charges of piracy, but Warren led a mob to free him. Warren and the mob then ceased Haskett and Walker and put them in prison. Elding, Warren, Graves, Lightwood, and other Bahamians held a triel and decided to depose Haskett and elect Lightwood as temporary governor. Haskett was shipped back to New York and then London. Haskett tried to place accusations of piracy on the lead Bahamians, but they never came to anything, and Haskett eventually escaped and fled to continental Europe.

But, at that same time, another war erupted in Europe, and England now faced Spain and France. So, in late 1703/early 1704, the Spanish and French burned out Nassau (the central settlement of the Bahamas) and everyone scattered. Graves happened to be in the Carolinas at the time, so he sailed with the new Governor that happened to be arriving in 1704 back to the Bahamas, where they helped any of the survivors that they could. The new governor never took up his office and left shortly after arriving. Many Bahamians outright left after the raids. For almost a decade and a half, there was no real government in the Bahamas. John Graves left for London, to petition the government to make the Bahamas into a royal colony (rather than one owned by a set of Lord Proprietors, which was the case at the time for the Bahamas, said Proprietors were pretty neglectful of the place). Unfortunately, no one really listened to Graves, and the Memorial publication was an attempt to promote his pleas to make the Bahamas a royal colony.

Eventually, in 1718, British authority returned to the Bahamas under Woodes Rogers. Sometime between 1707 and 1718, Graves had returned to the Bahamas, though he was advancing in age at this point. When Rogers started restoring the government to the Bahamas, Graves made claim to his old position of customs collector for the Bahamas, which Rogers gave to him, though Rogers complained of Graves' disabilities at the age he was at.

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That is all I have in my notes. Early Bahama history can be difficult to research, considering that in the late seventeenth and early eighteenth centuries, the population of all the islands fluctuated between only 500 and 2,000 people (not exactly a large body of people to leave behind paperwork), and the place witnessed so many raids and hurricanes that posed the potential to have destroyed documents that we could use today to study that period. We often have to rely on documents the Bahamians sent back to England and information garnered from other sources outside of the Bahamas. This period was a hard time for the Bahamas, so it's difficult to document it well (but not impossible, it just takes a lot of work).
As for the question on customs collecting, I never looked into that, and only know a little in passing based on other encounters. Custom Collectors had a significant responsibility, for from a maritime perspective, they had significant oversight on any ship traffic in a port/colony. They had to collect whatever dues a ship needed to pay. There were also many opportunities for graft and corruption. Are you a smuggler? Well, if you pay the customs man a bribe, maybe that shipment of illegal goods can just get by without being noticed, and other similar situations. I can imagine being the the customs collector for the Bahamas could be lucrative in the right hands. It was a port where illegal goods frequently flowed (when not burned to the ground by Spaniards and Frenchmen).