Nonsense has always been a condition of science, as has sense. The two form a dialectic at several different levels and in different ways.
It requires much nonsense to produce a little new sense. We are the inheritors of a long history; of thought reflecting on itself. One recalls the long apprenticeship of mathematics through Babylonia today as mere data, as lacking sense; with Greece adding sense - that is proof; the same goes for astrology - the divination of futures, the coronation of kings through the movement of stars that even today casts a long shadow as mere nonsense until Galileo picked up his telescope.
This is the dialectic of intuition and technique. Every beginning is intuitive, and every ending saturated with technique. Beginnings are playful, endings totalising. Towards the end a backward look at beginnings becomes scornful. One has grown up and ones face is set to the future - towards an end.
This - the attitude of Janus. We are divided creatures having the capacity to look both ways. Assuming the posture of reverence to the past, the future is discounted. Assuming the posture of rapture towards the future, the past becomes a kind of nonsense. Being embodied creatures we cannot transcend this condition; rather we transcend it only to find ourselves in a new landscape whose contours resemble the old (the Nietzschian return).
Take the Langlands conjectures in Number Theory. Their development was not through logic, but through intuition. A new vision born. This is the reverse of the dialectic of intuition & technique. Through a long apprenticeship of technique, a long lonely march in making the abstract cohere and become sensible & sensual in the Understanding, Langlands achieves enough clarity of vision to have a vision. One must not confuse the formula for the thought.
Abstraction is the name of a certain magical strategy in mathematics amongst others. But this isn't the condition of mathematics to mathematicians. Rather it is making the invisible Sensible to the Understanding and Sensual to the Imagination (the Kantian perspective). One could point to Conways The Sensual Quadratic Form.
Isn't language abstract - can you hold a word? Being one of the conditions of our existence & essence this abstraction is concrete. But Number being not such a condition, when spiritualised it is vision, and materialised it is measure. Through measure - the ladder - to spirit and to vision (Wittgenstein)
The doxa and data of Babylonian mathematics, at its time, then a beautiful vision, operating under a materially signifying star lost now; thus nonsense - now.
To turn towards your first question - about the 'increasing nonsense' of the contradiction between QM & GR. There are deep differences between the two theories. Physicists assume that the unity of the world as it presents itself to us should be reflected in their theories. This of course is a metaphysical assumption. This may be actually true, but on the epistemological level, the world may be arranged as such that we can never realise it - that is through our physical theories. Verlinde remarked as much in an interview.
The 'nonsense' produced, is then thought operating intuitively; headway is made in certain directions, other directions prove false; bridges are built, torn down and built again.
One might point to Ashtekars New Variables in as unblocking an old path in the quantising of GR; causal nets as a new thought; TQFTs placing Feynman path integrals on a formal basis as solid achievements amongst others. Gravity thought thermodynamically is a new direction theorised by Padmanubhan & Verlinde; one recalls here Bekensteins application of thermodynamics to Black Hole physics and Hawkings later assertion of it. Chris Isham, Doering & others are exploring QM in the context of topos theory which are a generalisation of set theory whose logic is intuitionisitic. Further they may or may not have 'points', and what points there are may have shape - this is getting away from Euclids dictat of a point being a breadthless. Toposes have been arranged in a series whose infinite incarnation gives logic shape. This is known as homotopy type theory. Urs Scheiber is using this to synthesise vast swathes of mathematical physics into one coherent & consistent system. One also notes the introduction of inconsistent & paraconsistent methods in mathematics with slow seepage across the border to physics.
Much more 'nonsense' will have to be produced before some sense can be distilled. The Topologist Alexsandrov said at the beginning of Topology that he felt threatened by the immense production of topological papers until he realised that they were 'nonsense' meaning containing ideas of little significance by moving away from the mainline of mathematics. But one should recall that ideas that move away from the mainline may find their proper context for their full expression much later - for example Lord Kelvins theorising of Knots as a model for atoms, reducing atoms to pure geometry; or Brouwer reacting against Hilberts championing of the formal and infinite (Cantors Paradise) by retreating to the intuitive and constructive.
One should recall that Physics is a tradition with roots going back to Antiquity, and before; and with a future no less long.
One doesn't need a deep understanding of Modern Physics to understand that the world is mysterious. This has been the condition of men at most times. Most thinkers at most times have remarked on it. Only in a civilisation saturated with technique is this familiar and widely understood observation become unfamiliar, strange and bizarre.
It takes only a little thought to see that knowledge is infinite, that we are finite, and thus what we know though adding up to a great deal for us, is always at the beginning of the infinite. The simplicity of this thought has become a platitude, and like all platitudes have suffered the indignity of being ignored. This does not make it wrong but to really know it, one has to experience it. This what William Blake, the English Poet & Artist meant when he wrote in the Song of Experience, the Marriage of Hell & Heaven:
If the doors of perception are cleansed, everything will appear to man as it is, infinite.
(1) are there simply limitations on what we may understand as meaningful and sensible because we lack the capacity to actually make sense of the results that we are getting?
Yes. When looked at the 'right way' this is always true. It is also very easy to forget this.
(2) what kind of theory do we end up with if we are prepared to simply settle for a theory that is completely sensible but within the bounds of the meaningful and sensible?
How can one work outside of the 'meaningful and sensible'? It is our horizon of thought as remarked by Wittgenstein in the Tractatus.
(3) does the abstraction of mathematics offer a via further up than experimentalists can even see/make sense of?
Yes; since mathematics is a tool, the more tools there are the better, the more efficient they are the better; but one shouldn't confuse mathematics for physical intuition, nor for thought or speculation. One should also recall that experimentalists do find things that are unpleasant suprises to the dreams of theorists - like the slow death of supersymmetry - or dark energy and matter. Remember experiment & theory form a dialectic. One shouldn't valorise one over the other. Human though it is.