With photons the physical attributes of the pattern are connected to the frequency/energy of the emitted light, which is a wave with wavelength in 400-700nm, right?
Yes, with photons the double-slit pattern is not surprising, since photons are excitations of a specific mode of radiation and the modes obey the usual classical wave equation, which of course shows a double slit pattern. (Of course, that changes if you mess with the modes by putting which-way detectors, which then get entangled with the field.)
With photons the contradiction comes because on top of this usual classical wave understanding they can also behave as particles (such as having only discrete detections with a sensitive enough photodiode), but to get to stuff that's really hard to explain with classical models you need to use things like Mandel dips; photons are not "particles", they're discrete excitations in the amount of energy in a mode of radiation, and it's mostly when you get to counting statistics (how often there is how much energy in the mode) that you step definitively outside of classical models.
With electrons and molecules is it (the pattern) not be related to the centripetal force exerted by the electrical respectively gravitational force applied on electrons/molecules by the edges of the slits (that have a certain crystalline structure)?
No. There is just no credible way to imagine that the electrical interactions between material particles and the slits will produce the observed behaviour, particularly when you require the pattern to maintain its shape but change its fringe width when you change the particles' energy and thereby their de Broglie wavelength. There's just no Newtonian models of interactions that have stood up to experiment here.
As to your other comments, refraction is a wave phenomenon but it does admit explanations using particle models (going as far back as Newton's Opticks, though not without their problems), but generally refraction is not thought of as a problematic phenomenon - it is perfectly well explained by the wave theory of light. The photoelectric effect, on the other hand, is indeed a strong piece of evidence of why light acts as a particle.
Finally,
Just because it contradicts math equations we are anyway unable to follow logically
is not a viewpoint I share. There is no requirement that Nature's laws be fully accessible to human understanding (though it would be nice). Quantum mechanics can be counterintuitive indeed, and it would be nice to have a theory that was more suited to our classically-honed intuition, but I don't think there's a requirement on nature to oblige.
Generally, the "math equations" is a bad way to put it: it is a solid theoretical framework with many experimental predictions and interconnections which have been verified in the real world, and there isn't anywhere near a suitable replacement to be seen. Quantum mechanics does have (rather deep) conceptual problems, most of them congregated around the measurement problem and the fact that we do not yet have a fully unambiguous and consistent way to set the postulates of QM in a manner that does not rely on an external distinction between what does and does not follow quantum mechanics (or what does and does not cause a wavefunction collapse). However, phrasing this as a hope that "one day physics will meet logic again" is very far from a constructive way to phrase things.
