1. The first question, namely, the physical meaning of \(R_{abcd}R^{abcd}\) was correctly pointed out in the other answer here, it appears in Gauss-Bonnet gravity.

2. But I cannot agree that gravity is not a gauge theory. Actually, there exists *many *approaches to GR as a gauge theory of the gravitational field. It started already with Utiyama in 1956, and then followed a crowd of people trying to formulate GR as a gauge theory, like Trautman, Kibble, Sciama, Feynman, Weinberg and Thirring. A complete list of references can be found here: https://arxiv.org/pdf/1210.3775v2.pdf

3. One cannot confuses a gauge theory as being necessarily a Yang-Mills theory. A Lagrangian of the form \(\mathcal L = \frac{1}{2}tr(D\omega\wedge \star D\omega)\) is typical of a Yang-Mills theory, but a gauge theory is a much more general framework. Y-M is a *kind* of gauge theory. To be a gauge theory, what one needs is a p-bundle where the potential \(\omega\) of the theory is a 1-form that takes its values on the p-bundle's Lie group, the field strength is the exterior covariant derivative \(\Omega = D \omega\) of that 1-form, namely, the associated curvature 2-form, and the Lagrangian of the theory being any combination of \(\omega\), \(\Omega\), \(\wedge\) and \(\star\), i.e., a 4-form \(\mathcal L = \mathcal L(\omega,D\omega)\in \bigwedge^4\mathcal M\), it is necessarily going to be gauge invariant, since nowhere we have introduced a local trivialization!

So how can we describe Einstein's GR in terms of a gauge theory? We choose our base manifold as spacetime \(\mathcal M\), as gauge group the group of general linear transformations \(GL(4,\mathbb R)\), acting on the frame bundle of \(\mathcal M\). The p-bundle of GR is therefore the bundle of frames \(F(\mathcal M)\) (the construction of the frame bundle and the action of \(GL\) is performed in all standard references, e.g., Choquet-Bruhat or Nakahara). Let \((e_a)\in \bigwedge^1\mathcal M\) be a tetrad in the spacetime (an orthonormal coframe), and \(R_{\mu \nu}\) be the curvature 2-form. Substituting Cartan's structure equation \(R_{\mu \nu} = d\omega_{\mu \nu}+\omega_\mu ^{.\alpha}\wedge \omega_{\alpha \nu}\) in the Einstein-Hilbert Lagrangian \(\mathcal L = \frac{1}{2} R\star1\), one shows (take this as an execise!) that the action of GR can be written like \(S=\int R_{ab}\wedge \star (e^a\wedge e^b)\). This is just the gauge theoretical reformulation of GR as the gauge theory with group \(GL\).

I recommend you to give a look in

http://www.iaea.org/inis/collection/NCLCollectionStore/_Public/12/619/12619799.pdf

https://www.amazon.com/Differential-Geometry-Cambridge-Monographs-Mathematical/dp/0521378214

https://www.amazon.com/Gauge-Theory-Variational-Principles-Physics/dp/0486445461/ref=pd_sbs_14_t_0?_encoding=UTF8&psc=1&refRID=5MP7Z8NM2TMZMTYK6GRE

https://arxiv.org/pdf/1204.3672v2.pdf