Okay, on t84 we had 3 cities, but had also settled one more and gifted to Darius as a TechSteal city. Darius and Sal were both pleased with us. Brennus was nowhere close to us yet. (Our Horse City was not within 9.5 tiles of his nearest city.) I haven't looked at your saves but clearly you need to have REXed faster.
Pretty much the same, we had 3 cities, also settled one more and gifted it (to Sal, who was the only AI with only 3 cities and willing to accept it at the time we were ready to gift it). Main difference was that we had built mids and gifted stone city after mids was completed. Our game up to that point was TGW->settle fish->settle stone->mids->gift stone->settle horse. Needing to finish mids delayed the gift so much that only Sal would take it anymore. We also wanted to gift it to Darius, but this was not an option. In the end gifting it to Sal turned out alright, he was the fastest techer early on and we got some good steals from there. Except that he also settled on our border before we even did any stealing, so gifting for stealing techs wouldn't have been necessary.
Under these circumstances, assuming we wanted the mids, I don't see how we could possibly have done anything to prevent the DoW from Darius. Okay, we could have gifted him a city before he started plotting, but it was not as he was a major threat and we didn't even know if he was on our own landmass. A city gift at that time would have been purely for tech stealing reasons, not security. It appears he started plotting at the end of t60, t61 or t62, don't have exact save. You gifted him a city t61, which for us could have been just in time to prevent him from plotting. Or not.
By the time the red fist showed up, getting him to pleased would have been of no use anymore. There was nothing to beg for without currency. As mentioned, we had no idea were he was, so we didn't even know in which direction to go to look for his incoming stack. In fact, we hadn't seen any AI culture or any AI units except the scouts in the beginning. As far as we knew, we could have been alone on our landmass. Also, you know the restrictions of the game, we had to keep most our warriors close to cap to be able to settle a new 3rd city after finishing mids and gifting stone city away.
So an honest question, what would you have done if Darius had showed a red fist start of t61, same turn you gifted him the city? Assume that you haven't seen any of his units to give away the general direction of his civilization, and assume that Darius has 4 potential targets that he is cautious with. Would the fist somehow have affected your plans and playing from that point on? We pretty much ignored the fist, which I still think was the right play. At least this was better play than to focus too many resources on trying to find Darius and any potential stack he has.
Our most questionable play was the mids. It cost a lot of resources to settle stone, lost several warriors to get the settler through the arctic barblands. Though the war also really ruined our chances to leverage Rep effectively. We wanted to grow Fish city, build library and run a couple of scientists. Instead we were emergency whipping walls and archers, while our workers had to run away and couldn't build any improvements. Rome also had to delay library a significant amount of time to build archers instead. But at least we put PS into good use later.
Edit: wow, you were lucky to get circumnavigation bonus t93! In our game Zara did that t72...