How big is the universe?

[Alex K]

Was the main reason for the inspiration of the many worlds hypo to try to logically explain the double slit/spooky distance?
Or was the idea justified to solve other dilemmas?

Yes, the double slit and all the related superposition experiments. But I don't tire to point out that theoretically, quantum superposition is directly tied to Schrödinger's wave equation and the wave function. Once you accept that Schrödinger's equation describes the world, you are automatically confronted with the question: what if I include not only that tiny atom over there, but also myself as the observer into the system which is described by Schrödinger's equation, what does it mean if I myself am in a superposition (which I will automatically be according the Schrödinger's equation if I observe an atom which is in a superposition).

If at this point you simply say: I accept that the wave function corresponds to something which really exists in nature, you automatically get Many Worlds, because all those superpositions of different possibilities are there in the wave function always, including you yourself being in one. It's only if you want to reject Many Worlds that you have to make the additional intellectual effort to explain why the wave function describes reality, but only very certain parts of it correspond to something that actually exists, or throw out any attempts to even say what superposition means and which things really exist (the pragmatic but philosophically unsatisfactory Copenhagen interpretation).

Do strings do anything to help our understanding of the double slit?

Not that I am aware. Quantum theory is kind of put on top of Superstring Theory (or vice versa depending on your perspective), but I am not aware that String Theory helps address the weirdness of Quantum Theory in any significant way.