I've been reading of the (surprising) fact we are uncertain on whether there is nuclear fission in the center of the Earth or not (yet we know so much detail on structures at the other end of the Universe), and I wonder, was Earth radioactive at its early stages? And if yes, would that affect abiogenesis in positive ways?
The neutrino flux from the uranium and thorium chains have been measured (by Borexino and KamLAND) as totalling around 24 TW thermal energy (out of a total around 44 TW). Potassium is unmeasured but significant.
These same measurements have strongly ruled out a central reactor model of the Earth (though I've read a arXiv posting proposing some valients for which I have not seen calculations).
Furthermore the natural radioactivity of the Earth typically contributes roughly one quarter of the basic background radiation dose that we are all getting all the time (mean is about 1 mrem per day). Potassium 40 is common in many building materials and in high potassium food such as bananas, carbon 14 is ubiquitous in the biosphere, and radon outgrassing can contribute more than cosmic rays in a improperly vented structure over granite formation (which can be so deep the residents aren't ware of them).
It is highly unlikely that there was any nuclear fission in the center of the Earth. For that you need a critical mass of fissile elements. These elements are actinides: uranium, thorium (and if you think of the really early Earth, then maybe a bit of primordial plutonium that has not decayed yet). The problem is that these elements do not reside in the core of the Earth. Instead, during planetary differentiation (when the entire Earth was molten and it split into the core and the mantle), these elements were dispersed in the mantle in very low concentrations, far away from what is required for nuclear fission.
That said, some processes in the crust can transport uranium or thorium preferentially. That's why we end up having ore deposits of these elements. Some isotopic signatures in Oklo, Gabon are consistent with natural fission of uranium. That is quite remarkable - that natural processes can concentrate uranium to a degree that fission occurs. Note that this did not happen in the center of the Earth, this was rather shallow on the crust.
Apart from fission, the Earth is radioactive, as mentioned in the other answers. The three most abundant heat producing elements at the moment in the Earth's crust are uranium, thorium and potassium. There's plenty of all three.
Yes, the Earth was radioactive. That's why there's hot inside.
Radiation should ionize atoms and dissolve molecules which might allow more chemical activity and more possible chemical compounds to be tried out, yet I don't think it could affect production of big and complex molecules, it could harm and dissolve them instead. I'm no expert though.