in the show it’s revelead that the Iron Bank invested in the slave trade of Essos. Would that be something that could happen in Book!Braavos?

Not directly, no. The First Law of Braavos forbids slavery, and that law is enforced:

“I know why the Sealord seized the Goodheart. She was carrying slaves. Hundreds of slaves, women and children, roped together in her hold.” Braavos had been founded by escaped slaves, and the slave trade was forbidden here.

“I know where the slaves came from. They were wildlings from Westeros, from a place called Hardhome. An old ruined place, accursed.” Old Nan had told her tales of Hardhome, back at Winterfell when she had still been Arya Stark. “After the big battle where the King-Beyond-the-Wall was killed, the wildlings ran away, and this woods witch said that if they went to Hardhome, ships would come and carry them away to someplace warm. But no ships came, except these two Lyseni pirates, Goodheart and Elephant, that had been driven north by a storm. They dropped anchor off Hardhome to make repairs, and saw the wildlings, but there were thousands and they didn’t have room for all of them, so they said they’d just take the women and the children. The wildlings had nothing to eat, so the men sent out their wives and daughters, but as soon as the ships were out to sea, the Lyseni drove them below and roped them up. They meant to sell them all in Lys. Only then they ran into another storm and the ships were parted. The Goodheart was so damaged her captain had no choice but to put in here…” (AFFC, The Blind Girl)

And that’s just domestically – internationally, Braavos has fought six wars with Pentos over the issue of slavery, and forced Pentos to abolish slavery and the slave trade. 

On the other hand, the Iron Bank of Braavos is quite active in lending to “Kings, princes, archons, triarchs, and merchants beyond count,” and they in turn likely buy and sell slaves (although I doubt that the Iron Bank would approve a business loan specifically to buy slaves as opposed to a general line of credit). 

However, we’ve had quite a few chapters set in Braavos, between Arya and Sam, and GRRM has not been sparing in the gossip, letting us know about merchants and nobles, courtesans and bravos, and that the Sealord is ailing and there will be a “choosing” soon, and we even know the name of one of the leading candidates. At no point in time have we heard anything about the Iron Bank supporting Volantis against Daenerys for interfering with the slave trade. 

My belief is that the slave trade nonsense was cooked up by Benioff and Weiss to give the Iron Bank a motivation to back Cersei against Daenerys because they’ve essentially given Aegon’s plot to Cersei (hence giving her the Golden Company) in order to give Lena Headey something to do. 

Are new Braavosi key-holders created? If not, how is the issue of extinct family lines counteracted?

Good question!

So here’s what we know about keyholders:

“Archmaester Matthar’s The Origins of the Iron Bank and Braavos provides one of the more detailed accounts of the bank’s history and dealings, so far as they can be discovered; the bank is famous for its discretion and its secrecy. Matthar recounts that the founders of the Iron Bank numbered three-and-twenty; sixteen men and seven women, each of whom possessed a key to bank’s great subterranean vaults. Their descendants, whose numbers now exceed one thousand, are known as keyholders to this day, though the keys they display proudly on formal occasions are now entirely ceremonial. Certain of the founding families of Braavos have declined over the centuries, and a few have lost their wealth entirely, yet even the meanest still cling to their keys and the honors that go with them.

The Iron Bank is not ruled by the keyholders alone, however. Some of the wealthiest and most powerful families in Braavos today are of more recent vintage, yet the heads of these houses own shares in the bank, sit on its secret councils, and have a voice in selecting the men who lead it. In Braavos, as many an outsider has observed, golden coins count for more than iron keys.”

In the first place, the extinction of family lines doesn’t seem to be a problem, given their 43-fold growth. In the second place, to the extent that it’s a problem it’s been solved by allowing the wealthy and powerful to buy shares of the bank’s stock, and thus becoming voting members. 

I’v never understood why the Braavosi underwriter that Arya kills had a hit taken out on him. Did he fail to honor an insurance contract?

Good question!

So here’s the relevant passage:

“He is writing each a binder. If their ships are lost in a storm or taken by pirates, he promises to pay them for the value of the vessel and all its contents.”

“Is it some kind of wager?”

“Of a sort. A wager every captain hopes to lose.”

“Yes, but if they win …" 

“… they lose their ships, oftimes their very lives. The seas are dangerous, and never more so than in autumn. No doubt many a captain sinking in a storm has taken some small solace in his binder back in Braavos, knowing that his widow and children will not want.” A sad smile touched his lips. “It is one thing to write such a binder, though, and another to make good on it." 

Emphasis mine there. The old man in question has been refusing to honor maritime insurance contracts – which the Braavosi call “binders” – and probably a widow or widows who’ve been screwed over “came to the House of Black and White and prayed for the god to take him.”

How does voting work in Braavos? Who can vote?

This is a tricky one, so bear with me: the answer is, we’re not entirely sure.

We know that the Sealords of Braavos are elected (for life) and we know a little about who elects them:

Neither prince nor king commands in Braavos, where the rule belongs to the Sealord, chosen by the city’s magisters and keyholders from amongst the citizenry by a process as convoluted as it is arcane. From his vast waterside palace, the Sealord commands a fleet of warships second to none and a mercantile fleet whose purple hulls and purple sails have become a common sight throughout the known world.

Part of what makes this complicated is that the electorate is composed both of magisters and keyholders, and it’s not clear whether all keyholders are magisters but not all magisters are keyholders or whether the two classes are completely separate. We do know that both magisters and keyholders form quasi-nobilities, with the Antaryons and Prestayns as two well-known houses of the former (the current sealord is an Antaryon, for example), and the Reyaans as one well-known family of the latter. Another complication is that the keyholders are officials of the Iron Bank of Braavos, which suggests something of the interconnection of state and corporation as was the case with the Dutch Republic and the Dutch East India Company

There’s also a suggestion, and this is more speculative, that there’s also a legislative body in addition to the Sealord’s executive. The books refer to a Hall of Truth (and sometimes a Palace of Truth, although it’s possible they could be two separate buildings) which is distinct from the Sealord’s Palace, where keyholders are “summoned to the Hall of Truth to vote.” While this could just be where the elections for the Sealord are held, the phrasing suggests otherwise (given that the choosing of the Sealord is a much more drawn out process than a sudden summons would suggest). Also, given how much Braavos loves its liberty, it would be surprising if they were less democratic than Lys, Myr, and Tyrosh. 

Are bravo blades closer to rapiers or estocs?

According to GRRM:

BRAAVOSI SWORDS
I recall Valyrian Steel fielding remarks on the sword’s hilt being cruciform, whereas some people have imagined that the true bravo’s sword is somehow different. I never had that impression, myself, but I admit to wondering if the bravo’s swords are supposed to have more encompassing hilts (swept hilts, bell cups, and those other sorts you see on rapiers)..

No, I vetoed all the basket hilts, bell cups, etc. Those are artifacts of a much later period in the real world, and would not be appropriate for a blade made by a castle smith in Westeros.

Would Braavos be in “control” of the headwaters/valley of the Upper Rhoyne, say the section north of Norvos, given their maritime nature? TY.

Unclear. Control probably goes too far – the Braavosi are pretty heavily invested in their maritime navy as opposed to a riverrine navy, and trying to control the Rhoyne would mean persistent conflict with Norvos, Qohor, and Volantis, on terms that would be pretty bad for the Braavosi. It would be a huge investment with a poor rate of return, when maritime control offers much better profit margins. 

Rather, I think the Braavosi are only interested in access to the Rhoyne. It gives them a much larger market they can sell into and buy from, it speeds up traffic for longer voyages, it offers them substantial military advantages while risking little of their own since the Sweetwater doesn’t link with the Rhoyne, etc.