There's a couple of things going on here. You raise some interesting issues about quantum mechanics, but you also make some crude mistakes about how QM and quantum computing work, and those crude mistakes severely limit the depth of debate that's possible on the interesting issues (and how seriously you can be taken on them).
I'll start with the plain-and-simple misconception:
- How can quantum computing handle with non-unitary operations like projective measurements causing the wave function reduction?
I'm not sure where you got the idea that "quantum computing cannot handle non-unitary operations like projective measurements", but it is 100% wrong. Quantum computing treats projective measurements as an integral part of the formalism: this is true in the circuit paradigm, where projective measurements are crucial for readout of the results, and it is even more essential in measurement-based quantum computing.
For more details, have a good read of an introductory textbook such as Nielsen and Chuang. If you want to raise meaningful objections to the validity of quantum mechanics and its constituent subfields, then you do need to understand what they do and do not say. The kind of misconception you've just blatantly espoused goes a huge way towards undermining any serious issues you might have raised.
Now, that said, there is a fundamental issue in the handling by quantum mechanics of nonlinear evolution, but this does not come in via projective measurements but rather through nonlinear unitaries which is where much of the real debate lies (beyond, that is, the measurement problem). The introduction of nonlinear evolution in quantum mechanics has a strong effect on quantum computing, and this extension of the theory is the object of serious study in the literature (example); anybody that says otherwise is either overstating their knowledge of the literature, or pushing an agenda.
That said, quantum computing as normally understood is strictly inscribed within standard quantum mechanics, in which unitary evolution is always linear. The only way that standard quantum computing would be able to interface with nonlinear unitary evolution is that it would fail to describe experiment - and indeed, if that happened, then quantum computing as we understand it would not be possible, as there would be situations where standard quantum mechanics would fail to be predictive. If this could be reliably and reproducibly probed, it would be marked as a breakthrough in the foundations of quantum mechanics of a caliber not seen since the 1930s.
More generally, though, quantum computing isn't a particularly good arena for testing the validity of quantum mechanics. Any beyond-QM effects that might be detectable via quantum computing would most likely be easier to detect via other means (such as quantum-driven precision metrology). But without knowing in which way the extensions go, it's impossible to say where they'd be easiest to detect.
Anyways, moving on.
Does quantum gravity make sense if the last shell is not quantum mechanical?
Yes? No? Maybe? Who knows! If the universe ultimately isn't quantum mechanical, and there is some underlying theory from which QM emerges, then the details of the situation would depend on what the underlying theory is. Some possibilities could include gravity inside quantum mechanics, and other possible theories would break the QM 'shell' before adding gravity. Without knowing which way the extension goes, it's impossible to tell.
And, finally,
I assume not lot of people would support the view that the ultimate theory is not quantum mechanical but classical, but quantum mechanics could be itself an effective theory
you assume incorrectly (example). There's a good number of people that would welcome the fact that there is an underlying classical theory from which QM emerges, and indeed there's been a lot of proposals for this over the past eight decades in the literature. However, there's the pesky matter that this theory needs to limit to QM, which isn't easy, and it needs to describe experiment. The reason people don't talk about this that much is that none of the proposed alterations have been very successful at producing a coherent picture of the world while fulfilling those two criteria.
