Kant is right that sensation is not knowledge. Mind is the active agent that creates all our knowledge. Kant's analysis that divides all data into the levels of sensation, perception and ideation is also correct.
Is the world a construct? I think that for each of us, what we regard as "the world" is a construct, as Kant claims. To me, you are my ideas about you. (See my article on
things.)
Is there such a thing as apriori knowledge? I don't think there is such a thing in the sense that Kant means. The newborn baby seems to have certain inbuilt perceptions, but not others, eg it does not recognise depth. It seems most unlikely that the newborn comes with Kant's twelve categories (or predicate logic). As for mathematics, it is not really apriori because experience of the world is needed before this can be used as the basis of abstractions such as number, sets and relationships. I must see two balls or two people before I can create the general concept of two-ness. Mathematics developed as a way of ordering and predicting the data of experience. Without that data there would be nothing for the abstractions of mathematics to arise from. Likewise, logic developed in order to control and predict things and processes in our world of events and objects. So both mathematics and logic are indirectly derived from experience, just as visual art is derived from the sense of sight: if we were all blind then no-one would paint pictures.
Kant argued that mathematical knowledge was both absolutely true and independent of experience. This view is untenable. The existence of the axiom of choice, which can be accepted as a true axiom or not, demonstrates that mathematics is neither necessary nor absolute. Godel proved that every consistent system such as mathematics will be able to express statements which will be neither provable nor disprovable within that system. At a deeper level, there are divergent schools of thought within the very foundations of mathematics.
The claim that science is absolute and its truth everlasting may have been persuasive in 1781, but to the modern mind, accustomed to scientific revolutions and the notion of falsifiability, it is unacceptable.
Is Kant describing the inherent and necessary operation of the mind? Animals too live in a world of objects, or so it would seem. It is hard to imagine what it would be like to live in a world not based on objects, eg a world with no stable objects but consisting of processes and flux. It seems safe to assume that all human beings live in a world of things, ie of relatively stable separate entities (level 2). As for the third level (ideas), there are probably cultures that don't use Kant's twelve categories. It is also arguable to what degree the categories Kant identified are central to our own thinking. One could also put forward other intellectual categories, not to mention emotional and subconscious factors, which no doubt mould our thinking more than we realise. In a sense Kant is talking psychology rather than philosophy.
If I were to define categories in the manner of Kant I would come up with something like: causality, excluded middle (A or not A), entities and relationships, induction, deduction, negation, persistence and repeatability, simplicity (Ockham's razor).
I agree with Kant that space is a mode of perception. It is the framework that we invariably impose on our perceptions of reality.
Time seems even more intangible than space. Certainly we cannot sense a moment or a second (nor can we sense an interval of space). I believe that the
flow of time is only perceptible to a mind, ie the flow of time is not present "out there" independently of observers. (See my article on
consciousness.) As for the past, it does not exist except as memories we have in the present, which we
interpret as referring to the past. So I agree with Kant that time is a framework deriving from consciousness.
If time and space are both subjective in the sense that Kant means, then what are the consequences for science? First though, what is science about? It is not the study of sensations. It is the study of objects and events, ie it starts on level 2. Science may be defined as all the theories and beliefs that are falsifiable. "Falsifiable" means that in principle a reality check of a prediction may negate the theory on which the prediction is based. What does this mean? It means verifiability at level 2 (things and events); ie science only talks about what we observe at level 2. Theoretical reasoning happens at level 3, and actual experiments occur at level 0 (things-in-themselves), but they are always sensed at level 1 (as stimuli), and translated to levels 2 and 3 by our minds when interpreted. Without interpretation they are of no value to science.
So it seems that Kant is right - science is about our conceptions of the world, not about the world itself.
What about Kant's contention that we cannot extend space and time beyond the limits of our experience? What would be the limit of applicability of space? We could limit it to the earth, or to the solar system, or to our galaxy. Such limits seem arbitrary. To set any particular limit on the applicability of the concept of space is just as paradoxical as to say that space is "real" (ie not subjective) but finite. In other words Kant has not removed the antinomy regarding space but merely recast it.
To give a modern answer to Kant's question of whether space is finite we need to look into general relativity theory and to decide whether the "curvature" of space due to gravity makes it finite or not. This is related to determining whether space is Euclidean or not.
What of the limits of time? Should we limit it to a human life-span, to recorded history, or the existence of homo sapiens? Again, any such limit seems arbitrary.
My view is that although space and time are modes of experience, it does not make sense to set limits to their applicability.
As for using the antinomies to limit the possible scope of our knowledge, I think that Kant's argument is specious. I cannot imagine the number 'i', the square root of -1, because it is not a quantity. For that matter, I cannot imagine pi either, but that does not render these numbers less valid than the numbers 0 and 1. (For that matter, can we imagine '0'?). The learning of mathematics forces us to extend our intuition. For instance we learn that infinity plus infinity is still infinity. (The number of odd counting numbers is infinite, as is the number of even counting numbers, as is the full set of counting numbers, consisting of both evens and odds.) In the same way as perceptual illusions do not invalidate perception, such counter-intuitive discoveries in mathematics or logic do not undermine reason. Rather than circumscribing the ambit of reason they show us how to extend it.
A more basic caveat is that Kant did not seem to be aware that the instrument he was using to critique reason was reason itself.
As to Kant's categorical imperative, it is a variant of the Golden Rule. Like all absolute moral precepts, it leads us into absurdity. Applied to lying it states that if you lie then you would be willing the whole universe to be dishonest; so you should not lie under any circumstances, not even to save your life.