Are there multiple Bitcoin markets?
Yes. They are almost entirely unregulated, so there is no 'order protection rule' that imposes consistency of prices across markets as US national exchanges do.
According to this question we should categorize Bitcoin as a currency
This is a matter for debate. It is not a government backed currency such as what can be traded in the Foreign Exchange market. It only has value based on what people are willing to pay for it, and what people are willing to accept. One has to be careful to understand that purchasing a Bitcoin is not an "investment" but rather a "speculative investment". A better speculative investment than buying a PowerBall ticket, but a speculative investment nevertheless. If Bitcoin goes out of fashion, it will go to zero. If it remains popular, it should increase in value.
What I don't understand is, why is it quoted as "BTCUSD"?
Ticker symbols vary across market data providers. While stocks typically only have one 'base currency' (e.g. IBM is usually always quoted in US dollars) some things are available in different currencies. As a result, market data providers may provide detail in the ticker to distinguish the different currency versions.
You can buy Bitcoin with different currencies, such as Euro, British Pounds, etc.
You might not know, but currency is not traded on an exchange but 'over the counter'. Some market data providers simply poll some large banks to see what prices they're willing to buy and sell currencies at, and take the best ones to come up with the rate. That is why it is hard to get details of volume and quantities traded - there is no central place where that data can be collected.
Most Bitcoin 'exchanges' are simply electronic platforms to perform over the counter trading (in a similar way to currencies). They are definitely not 'exchanges' as the Securities and Exchange Commission would define them (and be subject to regulation by them).
Some offer data like what you would receive from an exchange, like market data, trade volume, etc. But as they are minimally regulated. There are not a lot of mechanisms to ensure they're not just making the numbers up randomly. Not that this happens, just that there are not so many mechanisms to prevent it from happening.
At a given moment in time, does Bitcoin have a well-defined value in dollars, independently of where I buy it?
The price on one Bitcoin trading platform ('exchange') is usually close to the price on another. If the difference gets too large, people can 'arbitrage' the differences (buy on one, sell on the other, make a quick, risk free profit). Some tickers may get data one platform, average or take the best of a set. Ultimately the most valuable prices are ones where you are sure you can trade against them at the specified price - which would be the price from a single exchange. Otherwise it is an 'indicator'. As the value of Bitcoin can vary depending on people's moods in the market and there is nothing intrinsically valuable there is not much more to underpin Bitcoin's price, it is difficult to value it in any other way than what people are willing to pay for it.