Did John Calvin teach that fallen mankind has free will? The answer to that question might be a little more nuanced than one might suppose, and much depends on how one defines the concept of “free will” in the first place.
One thing at least is certain – Calvin was no fan of the term itself. He writes,
“[I]s it not ludicrous to hang so magnificent a title on so petty a thing? A fine freedom it is, when it is said that man is not compelled to serve sin, but is so much its willing slave that his will is held captive by the bonds of sin! I indeed loathe all quarrels about words, for they needlessly trouble the church. However, I think we should avoid using terms which appear rather absurd, particularly when they risk leading us astray.” (Institutes of the Christian Religion, p.47)
Some might indeed speak of “free will” primarily in order to affirm that sinners sin willingly, rather than by compulsion or against their will. But Calvin clearly held that this was not sufficient warrant to label such a thing as being “free.” He even calls such a term “absurd.”
More than that, as he himself states in the above quote, to use such a term runs the risk of “leading us astray” by causing someone to vainly imagine that “he is master both of his own judgment and his will, so that in his own strength he is able to turn one way or another?” (ibid).
Calvin concludes with the following remark:
“So if anyone would choose to use the word with a proper understanding of what it means, I have no further quarrel with him. But because I believe that it cannot be used without serious risk, and that to do away with it would greatly benefit the church, I would not wish to adopt it myself, and my advice, if it were asked, would be to give it up.” (p.49)
Wise advice. If one needs to explain and qualify a term every time it is used in order to avoid confusion and to prevent people from being led astray, perhaps it is best to simply not use the term at all.
Schreiberspace 
Free will is when a person decides for themselves what they “will” do, “free” of coercion or other undue influence. For example, after the Tsarnaev brothers bombed the Boston Marathon in 2013, they hijacked a car and forced the driver at gunpoint to assist them in their escape to New York, where they planned to set off the rest of their homemade bombs. The driver was not charged with “aiding and abetting”, because he was not acting of his own free will. But the surviving Tsarnaev brother was held responsible for his actions, because they were deliberate.
The concept of “free will” makes this useful and necessary distinction. Even in a world of perfectly reliable cause and effect, this distinction remains both useful and necessary to both moral and legal responsibility.
I still think Calvin’s advice is worth heeding when it comes to not using the term.
Fallen man outside of Christ is enslaved to sin, and indeed (as Paul says in Ephesians 2:1) is even dead in sin. As you say, this does not provide an excuse for sin, as no one acts unwillingly in their sins, nor are they coerced. But as Calvin says there, that’s not exactly worthy of the title “free.”
There is a delusion among scientists and philosophers that views reliable causation as a constraint upon our freedom. I presume Calvin has fallen into this same trap. But there is no such thing as “freedom from reliable cause and effect”, since without reliable cause and effect we could not reliably cause any effect, and thus would have no freedom to do anything at all.
So that definition is a bit silly. It is an “oxymoron”, a concept that contradicts itself.
Luckily, nearly everyone understands and correctly applies the simpler definition: “freedom from coercion or undue influence”. It requires nothing supernatural, it makes no assertions against reliable causation, and it is all that is required for both moral and legal responsibility.
What do you mean by “requires nothing supernatural”?
I mean that “free will” has a simple, pragmatic meaning, that works the same for everyone, regardless of their belief or disbelief in God.