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When we say "+3s/day" does this mean at the end of the week the watch has gained 21s?

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I would say yes with normal use. The thing is that normal use for one person might be different for another. My watches regardless of brand seem to vary in their accuracy depending how I use it.
 

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When we say "+3s/day" does this mean at the end of the week the watch has gained 21s?

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If, after 7 days, your watch has gained 21 seconds, then divide 21 by 7 to yield the daily rate of +3 seconds (21sec/7days = 3sec/day).

HTH
 

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Daily accuracy and Monthly accuracy are not strictly correlated.

If we are talking about a quartz movement, since it is usually very consistent (supposing the temperature variations aren't too big), then you can calculate the monthly accuracy multipliying the number of the days by the daily accuracy. So +0,5 seconds per day leads to an accuracy of +15 seconds per month.

On the other hand a mechanical movement depends on several factors, often not correlated among them, for example temperature and position. Therefore it is not so true the same calculation (monthly accuracy = daily accuracy x number of days). In fact these factors could change independently and affect the accuracy in opposite ways and you can have a daily accuracy always changing. In this case you could have one day with +5 seconds, another with -4 seconds resulting in a +1 second in 2 days.
 

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If a watch varies daily depending on it's position I think that the only way to get an idea of it's accuracy is to take its weekly variance and divide it by the number of days.
 
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