British English or US English?

Paris, here goes my take:

1) yes, it's an important part of the info one needs to do the best possible job;

• if source, not only do you have identical words meaning different things (eg to table a resolution) but you most certainly do have different cultural undertones that the interpreter should be able to identify and take into account, to be able to interpret properly
• if target, you do not want to use words, turns of phrase or idioms that may be misunderstood or misinterpreted by your audience, because they're from the "wrong" culture :-).. of course, audiences not being necessarily monolithic, you could find yourself "in a pretty pickle" ;-) unless going for the less-than-colourful bland common denominator...

2) I'd say consistency is always a good thing, including in language varieties :-) - that's accepting, for argument's sake, that one does have a choice ... and do not be influenced by Globish, it ain't a language! ;-)

3) ..importance and/or popularity are both subjective and depend on one's market, say in the EU British/Irish EN should be the norm, but of course a great many of those speaking and listening to EN are neither...

4) I'm afraid I don't understand, one speaks the EN one does, what exactly do you mean?