How do Protestants deal with pre-Reformation heresies?

By and large, Protestants follow Chalcedonian formulations in regards to Chalcedonian heresies.

Protestants by and large subscribe to hypostatic union, although our theories of communion vary widely from memorialism to consubstantiation. If you think in terms of the importance of these issues, however, the more "important" the doctrine (i.e. the nature of Christ is more important than the nature of the elements in the Eucharist), the more likely we are to agree.

Analogy to Common Law

The closest analogy would be like what happened in the Americas after we revolted from Great Britain. The UK had a long, established Common Law for crime, torts, etc... Those precedents in English Common law are still used in American Courts of Justice today. Why? Because it didn't matter if you had a President or a King - libel was still libel, theft was still theft, and double-parking was still rude. In the same way, old heresies are still heresies.

To stretch the analogy, however, what constitutes murder is almost identical. Libel, on the other hand, has far more nuance (and is much harder to prove) in the US than in the UK. Where does defamation become slander? Well, that's when you start getting lawyers involved.

If you ask any Christian outside of an obscure Dutch sect, "Is Jesus God?" You're going to get a yes. If you ask about Hypostatic Union, any priest or minister will agree. (And 90% of the people in the pew will say, um, sure, yeah.)

The point is, we agree on a lot.

What defines heresy?

Up until the rise of the Scholastic movement in the 1300s and the proto-reformers like John Wycliffe and Duns Scotus, there really was no such thing as a Protestant (or a Roman Catholic for that matter). There was simply "the Church." (I know someone's going to say "but what about the Orthodox?" They were part of "the Church" too, even if they were in disagreement about who was the head honcho.) Likewise, prior to 787 and the 7th Ecumenical Council, there was no Eastern and Western church - just the "church" (albeit with different Patriarchs). In regards to theological doctrine, there was (and still is) far more agreement than disagreement.

Look, for example, at the Nicene Creed. There is a heck of a lot of theology in that creed, and yet with the exception of one syllable at the end of filioque, 97% of the world's Christians can still agree on the entire summary! Each of the heresies noted by the Council of Chalcedon is a rejection of a tenet of that creed.

As such, most of the heresies as identified by the first seven ecumenical councils are regarded as heresies by modern Protestants as well. They are regarded as heresy by Orthodox and Roman Catholics too. There is a wide body of mainstream Christianity that really is a definable entity.

If someone tries to say that your salvation is dependent on a godly pastor, a Protestant can yell "Donatism!" as easily as a Catholic.

If someone says that Jesus was strictly God and not man, it doesn't matter if your RCC or Orthodox or Protestant, you can still say "that's Monophysite" and be right. (Incidently, a lot of Baptists are for all practical purposes Monophysite but don't know it.)

And to anyone who says that Jesus only appeared to be human or only appeared to die on a cross (like, say a Muslim), pretty much any Christian* can say "Docetism!"

Heresy simply means this - a theological position out of step with accepted majority thinking. Back in the days when people were persecuted for being out of step it meant something more, but nowadays, it just means whether or not the doctrine has historically been considered consensus or not. (Indeed, every heretic believed it was everybody else who was going to hell.)

How I learned to stop worrying and love the heresy bomb

To your original question of "What heresies are in the church today?" well, that's a book. The former dean of Virginia Theological Seminary once told me, "There are no new heresies, and there are no non-heretics. Everybody has a heresy, the question is simply which ones."

Trinitarianism and Christology are two classic examples of heresy breeding grounds. Between modalism and tritheism, monophysitism and monoergism and Nestorianism - there are so many errors on either side that the question isn't if one espouses a heresy, but which one. In practice, I'm sure I'm a heretic, but I at least can label my heresies :) The point is to note the error and seek to correct it in one own's theology (or point it out where there is a practical implication in others reasoning) rather than worry about or ignore it.

*Now - does that mean that there are some people who claim to Christian but hold to heresy?

Yup. Doctrinally speaking, Christians are not pure. We hold all sorts of mysteries in tension, and as such hold to heresy from time to time. By the measurement of Chalcedon, some groups are further away than others. It isn't a opinion, its a demonstrable fact. Historically, that's been the most widely accepted definition of Christianity ever.

Does that mean Chalcedon was 100% correct? That's when you start getting into bar-room philosophy and theological brawls. But if you just want to know what the (mainstream) Christian world calls heresy and what it doesn't, that's the place to start.