Would Calvinists classify themselves as determinists?

Kind of simple but I think the best way of understanding Calvinism with respect to God's immutable absolute free will and the tiny free will he gave men is like this:

The history of mankind was predetermined like a ship starting at London and arriving at New York. On the ship are puny humans like grasshoppers who can do whatever they want but they can't leave the ship or alter its course at all. They will always choose to stay on it. Eventually the ship will arrive in New York no matter how hard Devil, circumstance, nations, or whatever seems to be altering it otherwise, only to be achieving it.

In other words the envelope of history and all that we do within it are predetermined in that it is bound under his plan which can't be altered, but the many choices we make within that envelope is our true free will.

As regards Arminias I do not think he ever really understood Calvinism. Most of his writings are just doubts about what wise men thought.

With respect to Salvation by faith in Christ, it is included in the envelope, that is predetermined. So is our unavoidable arrival into heaven under his infallible protection. But it also involves choices on the ship, such as faith on Christ. Where the envelope and the activity on the ship cross paths, this is often referred to as the effectual call, in justification. For sanctification it is often referred to as 'the final perseverance of the saints'.

Where Calvinists differ from themselves is to what depth these matters are personally understood, and to what level they are hid in God and what level they can be explained. I think the more knowledgeable Calvinist will feel less can be explained than the novice. Life sort of works that way in general. Young men seek total certainty, old men seek heaven.

For Calvin himself, I think the answer is a qualified "yes" - qualified, because his views should be distinguished from a merely mechanistic view of what determinism means (as if the unfolding of time was just a physical system, nothing to do with God). Calvin saw the action of God at work in all things:

With regard to inanimate objects, again we must hold that though each is possessed of its peculiar properties, yet all of them exert their force only in so far as directed by the immediate hand of God. Hence they are merely instruments, into which God constantly infuses what energy he sees meet, and turns and converts to any purpose at his pleasure. [...] Every single year, month, and day, is regulated by a new and special providence of God.

And truly God claims omnipotence to himself, and would have us to acknowledge it: not the vain, indolent, slumbering omnipotence which sophists feign, but vigilant, efficacious, energetic, and ever active; not an omnipotence which may only act as a general principle of confused motion, as in ordering a stream to keep within the channel once prescribed to it, but one which is intent on individual and special movements. God is deemed omnipotent, not because he can act though he may cease or be idle, or because by a general instinct he continues the order of nature previously appointed; but because, governing heaven and earth by his providence, he so overrules all things that nothing happens without his counsel. (Institutes 1.16.2-3)

Events, which may appear to be random or contingent, are only perceived as such by us because we lack the perspective of God, and this includes "the contingency which depends on human will" (Institutes 1.16.8). The same is true about the future: "All future events being uncertain to us, seem in suspense as if ready to take either direction. Still, however, the impression remains seated in our hearts, that nothing will happen which the Lord has not provided." (1.16.9).

In his treatise The eternal predestination of God, Calvin explicitly denied that Adam in the Fall was exercising his will in contradiction to the divine will. Here, he makes a distinction between God as he appears to us, and God as he really is. His will (as we understand it) may be frustrated from time to time, but the true situation is that God's "decreeing will" or "overruling omnipotence" is always supreme. Now, Calvin does believe that Adam had a will! Adam's guilt comes from the way that his will was inclined in that moment, and the subsequent action (even though Calvin says he had no real power to will or act otherwise). The view he develops is essentially that our will is not truly a free will - it is not independent of God - though it seems real to us. We certainly do not achieve salvation on our own, nor even by joint action with God: it's all him.

There are several problems here which Calvin does not adequately resolve (and of which he is aware). He is taking an extreme position, and I think at some points he overreaches rhetorically. I think that the basic thrust of the Calvinistic attitude does not lead inexorably to this very strong view. In particular, there's a conflict between Calvin's view of God's sovereignty, and his view of our salvation, that hinges on whether we are truly morally responsible for evil that God "causes". He admits himself that he has no answer to this:

But how it was that God, by His foreknowledge and decree, ordained what should take place in Adam, and yet so ordained it without His being Himself in the least a participator of the fault, or being at all the author or the approver of the transgression; how this was, I repeat, is a secret manifestly far too deep to be penetrated by any stretch of human intellect. Herein, therefore, I am not ashamed to confess my utter ignorance. And far be it from anyone of the faithful to be ashamed to confess his ignorance of that which the Lord God has wholly enveloped in the blaze of His own inaccessible light!