Where are the inaccuracies in the Bohr model of the atom?
In hydrogen:
- It incorrectly predicts the number of states with given energy. This number can be seen through Zeeman splitting. In particular, it doesn't have the right angular momentum quantum numbers for each energy levels. Most obvious is the ground state, with has $\ell=0$ in Schrodinger's theory but $\ell=1$ in Bohr's theory.
- It doesn't hold well under perturbation theory. In particular, because of angular momentum degeneracies, the spin-orbit interaction is incorrect.
- It predicts a single "radius" for the electron rather than a probability density for the position of the electron.
What it does do well:
a. Correct energy spectrum for hydrogen (although completely wrong even for helium). In particular, one deduces the right value of the Rydberg constant.
b. The Bohr radii for various energy levels turn out to be the most probable values predicted by the Schrodinger solutions.
c. Also does a lot of chemistry stuff quite well (as suggested in the original question) but I'm not a chemist so can't praise the model for that.
This is an example of the "correspondence principle" in the broadest sense, that new theories should explain why old ones got some things right. The linked article discusses the Bohr model, but leaves some of your sub-questions unanswered. Going beyond that, how does an "electrons are somewhere specific" approximation lead to useful models of sharing and transferring electrons in covalent, ionic and metallic bonding? Well, we'll focus on covalent for now.
When physicists teach undergraduates enough quantum mechanics to do the hydrogen atom properly, electrons end up in specific atomic orbitals due to their quantum numbers, and each orbital can hold at most 2 electrons. The applications of Bohr-like reasoning you've brought up concern molecular orbitals, and these are a slightly more advanced topic; at this point I wish I knew what chemistry undergrads are taught about them, but I imagine Peter Atkins explains MOs with much the same rigour.
Like atomic orbitals, $\pi$ MOs hold at most 2 electrons (let's not get into $\sigma$ bonding for the moment). The Bohr lie would be that the electrons living in these orbitals have a precise location, and that orbitals form so as to get the electron count in each atom's outermost shell right and make for a stable molecule - you know, the usual $8$-electron rule (or $2$ for hydrogen, since it's trying to be like helium, not neon). The short answer to your question is that when we transition from quantum numbers for electrons in monatomic allotropes of an element to the analogous treatment of a molecule, the way the pattern of legal orbitals transforms is the same as would be expected on a classical model. Why? Because all you really need is the legal-number-combinations rules, not the way it's derived from the Schrödinger equation.
Let's consider the simplest possible example, $\mathrm{H}_2$. The simple model says, "we have one legal orbital, and it has room for $2$ electrons, which is just what we need, and they end up in an orbit like planets in a binary-star system". The more accurate model is, "again we fill the unique legal orbital with $2$ electrons, but the electrons' behaviour is quantum-mechanical". You can approximate the electrons in that orbital as two particles in a box (although that's not a perfect analogy), because they don't have enough energy to escape unless a photon excites them, nor can they fall into a lower orbital because those are full. With this constraint, the quantum effects are quantitative but don't make much of a qualitative difference.
The parallel between the Bohr picture and the Lewis diagrams isn't that great if you consider that the electron is moving in the Bohr model, while the electrons are static in a Lewis diagram.
If a Bohr electron was "at rest" outside a nucleus, as it is in a Lewis diagram or one of your organic-chemistry diagrams, it would immediately accelerate towards the nucleus. And I cannot see how you would modify a Lewis diagram so that the electrons were "shared" while still being in orbit around the nuclei.