There is no difference between "Ils voyaient bouger le cadavre." et "Ils voyaient le cadavre bouger"; there is no difference either in a multitude of cases that fit this model: "Elle sentait approcher les vacances.", "Elle sentait les vacances approcher.".

Surprisingly, you can say "Ils voyaient apparaitre un vieux paysan." et "Ils voyaient un vieux paysan apparaitre." More interesting is the fact that I feel quite uneasy too at the idea of using "Ils voyaient un vieux paysan paraitre." The sole analysis I can muster, out of my personal experience and without other references is the following: "paraitre" is a verb that denotes an action quite sudden and short; the tense of "voir" is the "imparfait", not a tense for the actions that occur at a point in time. Try another tense, the tense of actions in the past that happen at a point in time, that is the "passé simple": doesn’t now "Ils virent paraitre un vieux paysan." feel just as balanced as "Ils virent un vieux paysan paraitre."? In other words, according to a certain logic, which might be contested, you cannot say either of "Ils voyaient paraitre un vieux paysan;" or "Ils voyaient un vieux paysan paraitre.".
As to the second (Ils regardaient…) the verb "faire" is not as previously a verb for which an object is no possibility; there is the object "l’exercice"; notice the change if instead of "faire" with an object we use an other word, the word "faire" as a verb of substitution, without an object: "Les soldats avaient commencé à nettoyer la cour. Elle regardait faire les soldats.". We can also say "Elle regardait les soldats faire." or something more complicated as "Elle regardait les soldats faire, lorsqu’une troupe est arrivée au portail.". It appears that when the verb takes an object the inversion is not possible. It is probably be quite clear to you that in all cases when a pronoun is used instead of a noun there can’t be an inversion, but let’s look at some examples anyway:

• Ils les voyaient apparaitre.

• Ils les voyaient faire l’exercice.

• Ils te voyaient faire.

supplement

What I call "not at all sound" in a comment about my answer should be read "not very sound"; it bears upon a detail, important but of limited consequences. I join a few more example concerning a different point I made, so as to provide a means to better understanding.

• Le cocher brandit son fouet et le secoua violemment une fois; les enfants entendaient le claquement du fouet.

This shows the error "tense point-action" referred to in the answer. With "entendirent" the sentence becomes right.

We change the context a little:

• Le cocher secoua son fouet plusieurs fois violemment; les enfants entendaient le claquement du fouet.

Still wrong: "entendirent" once again makes things right; although the action is not a point action it is an action that has an end, whereas the "imparfait" (entendaient) is used for actions in the past that are not perfect, that is to say not perfect in the way of having no specified end and possibly no precise beginning that the context might indicate; what’s been called "point action" is a particular case of actions that have an end in the past.
Notice too that a "plus que parfait" could have been used: avaient entendu; this tense also is the mark of actions that have an end in the past; which one of the two to use is a matter of context (which might be already clear to you).

Another modification:

• Le cocher secouait son fouet violemment; les enfants entendaient le claquement du fouet.

Right; notice that "les claquements" could be used as well for "le claquement" without a change in meaning. One could say that "claquement", in "le claquement" and in this last example only, has a collective value.