The earliest known use of the word in writing in its present orthographic form was in 1528.
The document, MS Brasenose College, Oxford VII, contains the phrase, “fuckin Abbot” written in the margin. There are numerous other examples in the 1500s, but they were spelled really different (even from each other).
That’s according to the OED anyway.
So, yes, it was definitely around back then.
And the 1600 was when it began to really take on the meaning we associate with it today, of intercourse. Back in the 1500s its meaning was more along the lines of to fool, make fun of, or hit. Based on these date charts I would say that by the 1700s its transformation into the basic meaning it has today had pretty much solidified. The only uses that we have today that hadn’t been recorded by the 1700s are to betray or cheat, and as an expression of annoyance (ie “fuck this long-winded answer!”) and some of the noun uses (ie He’s a great fuck!).