In the same way "a thinking thing" is translated into Latin to res cogitans, how would you translate in Latin "a fighting thing" and "a running away thing"?
The core question seems to be finding good verbs for fighting and running away; then it remains to form their participles just like in your example cogitare > cogitans. I suggest taking a look at any of the several good online Latin dictionaries. I will give two suggestions here, but I invite you to look at their dictionary entries and see if they fit your needs:
Pugnare is a good general verb for fighting, giving the participle pugnans.
Fugere means running away, and there are many prefixed versions (au-, re-, per-) for different nuances. Its present participle is fugiens.
The exact choice of verbs and whether you should have res or a plain participle or something else depends on the context. If you can elaborate on how you want to use these translations, we can offer a more tailored response.
Different take, based on Wikipedia.
Fighting words are not fighting themselves but
are spoken words directed to the person of the hearer which would have a tendency to cause acts of violence by the person to whom, individually, the remark is addressed.
By analogy we can say that things do not think, fight or run away. Instead, they make a person or an animal do that. So, instead of a participle construction, I would go for a combination with causa:
fighting thing - causa pugnae
running away thing - causa fugae