1. In or on how many of the following items might you find a date, or dates?
a tree, a diary, a calender, an appointment book,
a newspaper, a box, a digital watch.
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2. There's a certain object that's the third of its kind. Yet strangely enough it's always referred to as the second of its kind. What is it?
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Wild guess here. Would it be the measurement of time? It is listed as hour/minute/second.

Only one of these not on the list that I can imagine is a tree...the rest have a date of some sort.

2. Would it be the "second hand" on a watch (an analogue watch, that is)?

Dates grow on palm trees. I think the only one without a date in/on it is a calender sic.

Yep, but a calendar does have a date. Many of them actually.

I suppose the paper a calender is smoothing could have some dates on it.

I realize I misread; it's used in papermaking, not printing. The machine might have the date of manufacture or something like that on it.

Answer 1 = 6 (all but the calender).

1. 6. All except a calender - "A machine in which cloth or paper is pressed by rollers to glaze or smooth it." (Oxford Dictionary)
I think that a date of manufacture on it is stretching things a bit, Tim. Perhaps I should have added "as part of its function".

2. Of course, it's the third hand of a watch, after the hour and the minute hands - referred to as "the second hand".

Thanks to all.