Well, it's traditional that instruments play in the same key. It's not god's law or anything, but it's the most common practice. They could play in different scales, it would just be a little tricky to avoid any dissonance, but then again, maybe a person wants that dissonance. (side note: *MANY* of the chords we consider "good" or consonant today, were once considered dissonant. 7ths, 9ths, 12ths, many of those chords were considered dissonant until some composer put them to good use)
If you want a track to sound "standard" and pretty much like most music you hear, then yes, you want the bass to play in the same key as everything else. But like I said, you could come up with other combinations. For starters, against the F Major, the bass could play in D minor. However, that's a disingenuous example, because F Major and D Minor are the same scale with a different starting point. But, since you're thinking of the Bass as being rooted at the D, you'll tend to want to resolve to that note, which will make things sound different. (I personally think Bass playing in relative minors sounds pretty cool)
Again, with counter melody it's the same deal. Most often, like 9.9 times out of 10, you'll find counter melodies to be in the same key. Not a concrete rule, though, it's been broken many times by many different musicians and composers. However, trying to write counterpoint in different keys seems like it would be waaay tricky, but it yields interesting results. One popular trick is to have all ascending melodies play in the natural minor, and all descending melodies play in the harmonic minor. Little things like that are somewhat common. The scales are still in the "same key" but small intervals of the scale are different between the two instruments/melodies.
