In what time period is "Romeo and Juliet" set? When was it written?

Well, we don't really know for certain when "Romeo and Juliet" is set, or when it was written either. Scholars currently think that the play might plausibly be dated to 1595. We know too that it can't be any later than 1596, or any earlier than 1591. 

There's nothing in the play that would plausibly date it as a medieval play, any more than a play set in the 1300s and 1400s. The feasts, the weapons, the street brawls, the tomb - all are perfectly plausible as a "modern dress" play for Shakespeare: though, of course, one set in Verona, a hugely fashionable city which, although most Elizabethans would never have visited, was the subject of some fascination.

Shakespeare based his play on a poem by Arthur Brooke, and another (Italian) version of the same story by Luigi da Porto. da Porto sets the plot in modern day Verona - 1530, when he wrote the poem. I think the only real assumption that we can make about the play's setting is that Shakespeare's actors (as the evidence suggests they did) would have played it in the modern dress of the day.