ELI5: What is the effect of ultrafine Teflon (PTFE) particles in rats and mice
I think the authors' choice of words might have confused you a bit.

We found that neither the ultrafine Teflon particles alone when generated in argon nor the Teflon fume gas-phase constituents when generated in air were toxic after 25 minutes of exposure.

What the authors mean is that when they used only ultrafine Teflon particles on the rats it did not kill them, nor did the Teflon fume gas-phase constituents when the rats were breathing that air for 25 minutes.

Only the combination of both phases when generated in air caused high toxicity, suggesting the existence of either radicals on the particle surface or a carrier mechanism of the ultrafine particles for adsorbed gas-phase compounds.

The individual substances, that is, ultrafine Teflon particles and Teflon fume gas-phase constituents were non-toxic for the rats. When the two were combined together in the air and the rats were made to breathe in that air was the point that they died, meaning that when these substances were combined in the air, some radicals$^{[1]}$
were being produced or some form of carrier mechanism was taking place.

The Teflon particles alone, I think means the compound before they heat it.

No, they meant that the ultrafine Teflon particles after heating. Alone meant these Teflon particles, by themselves, are not toxic.

what is the 'fume gas-phase when generated in air'

Gas-phase is simply a way of referring to the gaseous physical state. See here$^{[2]}$ for the definition. The phrase fume gas-phase refers to the fumes that the author observed on heating the Teflon to the temperature at which it turned into gaseous phase (phase is the word for referring to the physical state of a substance) while producing fumes.

How would coagulation affect the toxicity of the fumes? Does it change the chemical structure of what the animals inhale somehow making it less toxic? Naturally speaking, I would assume 'more' particles would result in higher levels of toxicity, not less.

Its not about more or less here. After some time, coagulation takes place causing the Teflon to take a solid form in the whichever utensil the researchers were using. What they mean by the coagulated Teflon being non-toxic is that in whichever way they made the animals (they have not specified which animal though) ingest it, it did not kill them. The quantity of the Teflon, the form in which it was ingested and by which animal was it ingested has not been specified, at least I couldn't find any clear reference in the abstract and I don't know how to access the full paper.
References:-

IUPAC. Compendium of Chemical Terminology, 2nd ed. (the &quot;Gold Book&quot;). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). Online version (2019-) created by S. J. Chalk. ISBN 0-9678550-9-8. https://doi.org/10.1351/goldbook.
IUPAC. Compendium of Chemical Terminology, 2nd ed. (the &quot;Gold Book&quot;). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). Online version (2019-) created by S. J. Chalk. ISBN 0-9678550-9-8. https://doi.org/10.1351/goldbook.