How widespread is panentheism within Protestant schools of thought?

This is a fantastic question. Here's my answer.

Firstly, I would argue that the definition you use from Oxford English Dictionary is actually not as encompassing of the actual theology as it could be. Going against Oxford English Dictionary is a bold claim, so I'll provide another definition.

Stanford Encyclopedia of Philosophy

“Panentheism” is a constructed word composed of the English equivalents of the Greek terms “pan”, meaning all, “en”, meaning in, and “theism”, meaning God. Panentheism considers God and the world to be inter-related with the world being in God and God being in the world. It offers an increasingly popular alternative to both traditional theism and pantheism. Panentheism seeks to avoid either isolating God from the world as traditional theism often does or identifying God with the world as pantheism does. Traditional theistic systems emphasize the difference between God and the world while panentheism stresses God's active presence in the world. Pantheism emphasizes God's presence in the world but panentheism maintains the identity and significance of the non-divine.

With this more elaborate denotation we can move on to the questions.

1. What does panentheism imply that a classical Christian formulation of omnipresence does not?

Panentheism implies that creation itself is 'in' God, and that God is 'in' creation. The latter phraseology does not seem very threatening to classical theists. So it is that Thomas Aquinas writes the following:

God is in all things; not, indeed, as part of their essence, nor as an accident, but as an agent is present to that upon which it works. For an agent must be joined to that wherein it acts immediately and touch it by its power.

Aquinas, who is often regarded as the greatest classical theistic thinker of all time, makes note of the difference between two understandings of how God is present in all things; though Aquinas does not name them, it is clear to the reader that Aquinas is comparing panentheism and omnipresence. While God is 'in' all things, God is not in the essence of the said things, but rather is present to them, so as to ground their own essence in His essence of 'being'. This is very complicated speech for one unfamiliar with Aristotelian language.

To simplify the argument, think about moving a ball. The essence of the ball's motion is grounded in your moving it, but in some capacity there is a distinction between yourself and the moving ball as regards the nature of movement in both. You are not 'in' the motion of the ball; your type of 'motion' is the independent source of the moving ball. This analogy is primitive and should not be too deeply analyzed, but it is somewhat a more simple way of looking at the panentheism-omnipresence discussion. Panentheism holds that God is in the reality of our being, as a part of our essence. It holds that your movement is 'in' the ball, as a part of its own essence. Omnipresence holds that God is present to creation in order to sustain their independent essences of 'being'. It holds that your movement is distinct from the ball's movement, and that the ball's movement is due to your presence, that is by necessity outside of itself entirely, but also 'present' to and 'in' the ball. God in His entirety is not within and bound to His creation anymore than you are within the ball; God is present to the creation, just as you are present to the ball. Omnipresence does not imply that God is within everything's essence; only that He is present to everything's essence, sustaining its existence.

Although panentheism might seem the more 'mystical' and 'complicated' view of God, it holds poorly against the far more mysterious doctrine of omnipresence. Panentheism can be dismissed in a rather short paragraph. Omnipresence requires far more extensive understanding of the universe and God.

2. How widespread is panentheism within Protestantism?

The most predominant denomination that I would say actually supports panentheism (the real deal, not Orthodox or Roman Catholic 'mysticism') is the Unitarian Church. As of 2012 there were about 343,000 adherents to Unitarian Universalism worldwide. The problem with noticing panentheism's influence is in the obscurity of its meaning; it is very closely associated with omnipresence and many people who state that God is 'in' creation typically do not mean it in the way panentheism is implying. As stated above, I don't buy into attempts to think that many Orthodox and Catholic thinkers hold to panentheism. The result of people believing such is likely selective reading and a misinterpretation of poetical speech.

3. What problems do Protestants who reject basic panentheism have with it? By basic panentheism, I mean a panentheism that isn't attached to open theism and the other problematic doctrines mentioned above.

The problem with panentheism as a philosophy remains the same in its most basic sense without any additional interpretations or doctrines added to it. It is this same problem that most Protestants understand if they reject panentheism. Famous Protestant theologian William Lane Craig thus writes the following:

It should go without saying that the justification for panentheism cannot be biblical revelation or teaching. For the Bible knows nothing of the doctrine that the world is part of God.