is it possible that the USD be strong against a currency and weak against another one at the same time?
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Yes it can, and usually is.
Each currency has its own particular fundamentals underlying its own supply and demand. With some currencies these will be more powerful than those affecting the USD and with others these will be less powerful. Some fundamentals are positive and some are negative.
This is the basis of trading strong/weak analysis. I.e. identify the strongest and the weakest against a common benchmark and trade that pair. All the other currencies will lie somewhere between these two extremes, and the USD is no exception in this relative positioning. Sometimes it will be at one extreme or the other and at other times it will be in the middle somewhere.
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Yes it is. The US Dollar is not the strongest currency in the world. It is weak against the British Pound while strong against the Euro. The strength and weakness of the currency depends upon its supply and demand and is not a representative of a stronger or weaker economy.
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It seems to me that the dollar will have a massive impact on the market for a long time to come anyway - it always has a relatively stable position.
On a fundamental level, the US dollar is displayed strongly.
And why is that?
News, reports, a huge number of events.