I'm not an enzymologist in any way (and certainly not a clinical one), but hopefully these answers will make sense, and won't be (too) wrong.
1) Can enzymes be concisely defined as catalysts with complex make-up that makes them highly specific (selective) in the reactions they cause?
Yes, they're catalysts with a complex make-up (depending on what that means exactly, and depending on the specific enzyme too), but they're not necessarily highly specific. "Enzyme promiscuity" - the ability to react with diverse substrates (other molecules which constitute the targets of the enzyme) - is a big thing. Sometimes this happens because the conformational state of the enzyme has been (reversibly) changed, and therefore its specificity is biased towards one substrate and away from the other. In other cases, the same enzymatic state can bind to different substrates, although the affinities might be substantially different for each.
2) Do most enzymes catalyse in a rate only limited by their ability to diffuse in that which they are reacting with?
Staying with a simple model, let us define an enzymatic reaction the set of the following sub-reactions:
1 - The reaction where the enzyme meets and binds the target, which may be accompanied by the opposite reaction (dissociation from the target before completing the catalysis);
2- The actual catalytic reaction which transforms the substrate into a different molecule.
The (global) reaction rate depends therefore on the rates of association and dissociation of enzyme and substrate, on the rate of the catalytic reaction, on the concentrations of enzyme and substrate(s), and on the concentration of possible competing enzymes. These rates are population averages (i.e., averaged across many molecules. The timing of the reactions at the single molecule scale is random). At this level of detail, you can just imagine the the diffusion rate is lumped into the reaction rates.
Assuming all else is equal (that, in your words, the ability to diffuse is the same), the relative numbers of enzymes and substrates will change the global reaction rate though. Production will saturate for high amounts of substrate. Intuitively you can think of it this way: if all the enzymes are busy doing their good catalytic work, then they're blind to the addition of more work, because they're already operating at their maximum speed and have no free hand to spare.
The picture gets a bit more complicated if you put them in a crowded environment full of other proteins, and in very small volumes. The output of an enzymatic reaction can be substantially different in those conditions (both quantitatively and qualitatively), and this is an active area of research at present.
3) I read that some enzymes at least appear to catalyse even faster than it takes them to diffuse in what they react with. On the surface this sounds like a paradox, cause in effect it would mean that parts of the enzyme or its reactive core sort of reach the entire field of the reaction before this happens for the entirety of the enzyme.
Not sure what this means. Where did you read that and what is the exact quote? Enzymes are molecules (or form a part of larger molecular complexes) that move around in a random fashion or get transported by other types of molecules. They interact "physically" with their targets through a region of the enzyme's "surface" called the active site. This is where the catalytic reaction takes place. From this point, while the speed of the catalytic reaction(s) appears to be typically slower than the speed of the earlier binding reaction(s), that needn't be the case in some solutions. There is no part of the enzyme that reaches the "field of the reaction" before the rest of the enzyme: there is one bit of the enzyme that is active in the reaction, but is always attached to the rest of the molecule.
You also need to be careful when translating enzymology jargon into a mental picture: such jargon is usually a description of the output of an ensemble of molecules, not of an individual molecule.
and last, but certainly not least:
4) If an enzyme diffuses very fast in a distinct volume it reacts with, how will this possibly be a predetermined factor for it to not extend its effects in nearby (but not in the immediate one) volumes? In other words: is this a safeguard for even more specialised use of the enzyme?
If an enzyme diffuses very fast it will spread faster to nearby volumes, unless there is some sort of barrier (and often there is). There are mechanisms to maintain enzyme "specialisation" in a given moment or process though. For example, they can be sequestered by other molecules, or they can be activated and inactivated by other proteins or smaller chemicals (by either blocking the active site or changing the 3D structure of the enzyme (e.g., by changing its distribution of electrical charges), such that the enzyme can't recognise its target anymore).
