Brussels seems to be the dilema in that although capital of Flanders, it's overwhelmingly Francophone. Does it have a lot of French-speaking indigenous Flemish for some reason?
Yes and No. Many Francophone inhabitants of Brussels and their suburbs have Flemish roots, their (grand)parents spoke the Brussels' (or some other town's) dialect of Dutch, although they usually never learnt 'standard' Dutch. Originally, Brussels is a Dutch-speaking city.
The (Francophone) Belgian elite saw the Dutch language as a 'peasant language' and tried after Belgian independence to 'Francophonify' the Dutch-speaking part of the population, essentially by giving them an inferiority complex. For example, Dutch-speaking parents speaking only French with their children and sending them to Francophone schools to give them a better future.
The Francophones were especially successful in the city of Brussels, where there was also an influx of Walloon civil servants and later of EU civil servants and North African immigrants. Usually, Francophone Flemish Belgians consider themselves Francophone, not Flemish.
Only since the beginning of the 20th century a counter-movement from Flemish side started, which finally culminated in the division of the country into all kinds of communities and regions, all with their own governments and parliaments. Also, more people from Brussels are bilingual French/Dutch now than in the last decades.
Nope. And why should they? Apart from Limburg province, where the border was different, Flanders and The Netherlands are separated since the 16th century, only to be re-united in the United Kingdom of the Netherlands (1815-1830).
Oh yes, they have 6 million inhabitants, when excluding Brussels. More than for example Norway, Finland, Ireland or Denmark. Since the Flemish economy is performing well, viability shouldn't be a problem.
Good question. Although originally the Belgian monarchy was more popular in Catholic Flanders than Socialist Wallonia, now the situation is reversed. Although King Albert is quite popular, as was his late brother King Boudewijn, Albert's children are not. They speak Dutch with a strong French accent and many doubt if Crown Prince Filip has the intellectual capabilities for becoming King. Some scandals concerning Filip's younger brother Laurent didn't help. So, a Flanders Republic isn't unthinkable.